Conservation of energy, substance dualist and free will

In physics there is no place for anything like "unphysical substances", just like there is no place for anything "supernatural" or "unnatural". Anything we can perceive with our normal senses (or with thoroughly understood extensions of our senses, such as various kinds of microscopes, telescopes, etc etc), in a controlled way, repeatedly, and agreed upon by people irrespective of their cultural or religious upbringing, can count as "empirical" evidence.

However, your argument that anything acting on neurons in a brain would be "adding energy" to the universe doesn't really seem sound. It's simpler than that: If something (a soul, mind, ghost, whatever someone might call it) would indeed not be physical, then there is no conceivable way it could act on matter. It's not just so that there is no conceivable way in our current physics, there is no way to conceive of this, to explain this at all: no dualist has ever managed to come up with any satisfying explanation. This embarrassment is very obvious in Descartes' writings (about the pineal gland) and it was probably the main reason for Spinoza to reject Descartes' dualism and opt for a monistic view in which mind and matter are two "aspects" or "attributes" of the very same physical process (or substance): extension and thought. Spinoza's two attributes may also not be entirely satisfactory, but any type of substance dualism will have a problem explaining how the two substances can interact.

Now, dualists counter with the objection: "You, physicalists have also never explained how mind (thought, experience, consciousness, qualia) arises out of pure matter." Indeed, there is still no complete theory of mental processes. But there are lots of simple and more complex phenomena that suggest that no mind can exist independent of matter. For instance, when someone falls on their head (hard), they... lose consciousness. Why would they need to lose consciousness if their minds are independent of what happens with the body? Everytime I sit in a dentist chair, I silently thank the pharmaceutical chemists who developed lidocaine. Surprise! You block some local sodium channels in nerve cells and suddenly you feel no pain! How would this be possible at all if the mind was not fully embodied in the brain and central nervous system? There are just too many purely empirical facts similar to this that all make dualism absurd.

"Ah," the dualist counters, "but there are Near-Death-Out-of-the-Body experiences! That proves the mind is independent!" Not at all. Our experience of self-localization (of being situated in our bodies) is itself constructed by our brain. We normally are not aware of this, but this can be shown by controlled psychological experiments. This experience can be manipulated in pretty uncanny ways, making it appear as if one is located outside of one's actual body, as shown for instance in the research of Olaf Blanke. (Some of his research papers are very accessible, also without extensive background knowledge of neurology. Wikipedia has quite a few interesting references.)

It is false that "if X is causally efficacious, then X does work", so all of the difficult to nail down parts of the question (like what it means for something to be physical/unphysical, why souls shouldn't have associated energies, etc) don't actually matter.

For example:

A static magnetic field does no work on a moving charged particle, but it exerts a force and changes its velocity.

Gravity does no work on a body in a circular obit, but is obviously the cause of an effect.

Constraint forces in general (for example, the centripetal force keeping the cars of a roller coaster from flying off) do no work, but are obviously the causes of effects.

You ask:

If there is unphysical substance that exercises free will without being affected by prior physical causes then it does work on the neurons that changes their trajectories which means more energy is added to the physical universe. However if the energy of the universe is constant according to the first law of thermodynamics, doesn't the existence of such an acting nonphysical substance contradict the first law of thermodynamics?

While it is true that great thinkers such as Rene Descartes believed in substance dualism, modern physics rejects such notions as mental substance. Therefore, when you try to discuss mental substance within the framework of thermodynamics, you are attempting something which violates a basic presupposition of physics and physicalism. Of course, there are several famous philosophers who might have a stance on this, including Sir John Eccles.

What you are asking about is called mental causation (SEP) and is widely rejected by modern physicists because of a lack of empirical evidence or sound theory of mechanism of interaction. There are those physicists who resort to quantum mechanics as a potential refuge for the possibility of such an interaction, but no theory has emerged as a widely-held contender. As such, any such theory is largely metaphysical speculation at this point, and no canonical answer has emerged to answer your question. Epiphenomenalism is another possible corrective response to your question.

This is on Philosophy.SE, so I hate to give a physics answer, but I think the physics based answer is valid none the less.

Energy is not conserved in general. Energy is conserved in a closed system. Energy does not need to be conserved in an open system; in fact, we define a closed system or an open system based on whether energy is conserved. In practice, there's a lot of very interesting systems which are well modeled as closed systems, even though they are technically always open with their environment. This makes the concept of a closed system, and the conservation of energy, valuable to physicists (and engineers, who leverage the ideas).

Energy is not conserved on the Earth. There's energy constantly entering that system in the form of sunlight. To first order, energy is conserved by the Solar system, as that sunlight was created by nuclear energy from the sun, within the system. The universe as a whole is sometimes thought of as a closed system, but if you get into the study of that, cosmology, you find that we add a whole lot of really interesting terms to the equations to make our model align with the reality we observe with our telescopes. In a very handwavey sense, that's where the search for dark matter and dark energy goes today: we see the effect of something that sure looks like matter or energy, but we don't fully understand the physics of how it works.

The key distinction for this topic is that, in an open system, we have to make statements about the boundary conditions - how does energy move in and out of the system. The details of that boundary cannot be derived from the behavior of the system internally. For this question, this means that the potential energy transfer between the non-physical and physical cannot be gleaned from simply observing the physical system.

It could be argued that said interaction cannot add energy to the system on average. Perhaps for every joule of energy added by the non-physical, it must remove a joule later. That would maintain the conservation of energy on average. However, it could not be explained as being because of the conservation of energy, as conservation of energy in physics only applies inside the system, not on the boundaries. It would need to be explained by some other metaphysical rationale. We would have to say the effect of this is conservation of energy, rather than claiming that conservation of energy is the cause.

The interesting question for a physical scientists would immediately be whether that energy can be harvested by constructing an appropriate physical machine. If it could, that would drive the scientist to try to understand the properties of that boundary, and perhaps expand our definition of physics further. If it could not, then it would leave open a question as to whether the non-physical substance is actually physical, or if it is merely physical material that is not yet understood. It would not dismiss the position of the substance dualist, it would merely be incapable of proving their position for them.

There is plenty of energy available in the brain. There is no need to "add new energy". Naturally no laws of physics are violated, the being uses the energy it has to do some physical actions.

From the philosophers standpoint the being is just exercising its biological ability to control the flow of energy in its body.

The notion of what constitutes energy has been revised several times as science discovered apparent violations of the first law of thermodynamics. Each time they were able to include the new form of energy in a way that left the conservation of energy true.

So one way to deal with this issue is by positing "nervous energy" which is basically a form of potential energy that minds can store up. Minds can effect the physical world either by taking energy out of the physical world and storing it as nervous energy or by using up nervous energy, putting energy back into the physical world.

However another possibility is to simply deny that conservation of energy holds in interactions between mind and matter. The conservation of energy is not sacrosanct in physics. There are various consequences of General Relativity that supposedly violate the conservation of energy. And there have been other theories proposed that violate conservation, such as a theories of a steady-state universe that involve the constant creation of mass or energy. The point of this is that it is not "unscientific" to suggest that conservation of energy may not hold under certain circumstances.

This is particularly true since all of the laws of physics were developed by using experiments that did their best to minimize the effect of the experimenter on the experiment. In other words, physics is the science of matter in the absence of mind. It is a bit problematic to take results that specifically exclude mind and to assume that those results also apply when mind is involved. Physics has run into this issue several times where expanding the domain of enquiry completely changes the underlying theories.

I'll also respond to this quote from mudskipper's answer:

If something (a soul, mind, ghost, whatever someone might call it) would indeed not be physical, then there is no conceivable way it could act on matter. It's not just so that there is no conceivable way in our current physics, there is no way to conceive of this, to explain this at all

This is clearly false. The idea that mind can directly control matter is so natural and conceivable that it is a common trope of fiction. Yoda making the starship rise out of the muck, or Jean Gray raising an aircraft to save the team. Perhaps what mudskipper means as that we can't conceive of a mechanism behind such a thing, but mechanisms are physical, not mental. When a philosopher draws a conclusion from a couple of premises, there is no need to provide a mechanism to explain how he did so; he did so because it was logical. Asking for a mental mechanism is like asking the color of concepts--it makes no sense.

The requirement for a mechanism is on the other side. If you deny the existence of a non-physical mind, then you have to provide a physical mechanism that would lead to mental experience, and that is the real problem that has no answer. There has been no progress made on that question ever, since it first came up.

See much past discussion of dualism. It does not approach being well enough defined to discuss mechanisms of interaction, conservative or not.