First law of thermodynamics, the conservation of energy, doesn't this law all but guarantee that regardless of how far the universe expands it will forever contain its original amount of energy? All the matter in the universe will break down and all the energy will reach an equilibrium. However, because of the conservation of energy, the universe will never reach absolute zero will it? As it expands, the universe will continually get closer and closer to absolute zero, but it will never completely get there, or will it? How many billion years will the universe be between 1.0 degrees Kelvin and .01 degrees Kelvin. Is their a mathematical formula that describes the cooling rate of the universe towards absolute zero, once the universe has reached thermal equilibrium?
2 Answers
Your general reasoning is correct. However, one must keep in mind that all of this is based on models of the universe and the concept of the conservation of energy does not always hold true on a universal scale within all models.
Even more important to keep in mind is that often what we refer to as an observation is really a model-based observation. In other words an outcome that is justified by viewing the observation from within the parameters of a particular model. In the case of the expansion of the universe this is most often the Friedmann-Lemaitre-Robertson-Walker (FLRW) model. This raises the question of what do we actual know about the expansion of the universe.
Let me give you an example. When a group of astronomers state that they have measured a red-shift in light from a far-off galaxy as evidence of the expansion of the universe, one must note that their conclusion is based on interpreting that observation within the framework of the FLRW model (or some other model).
The actual observation is that light from a far-off galaxy is seen with light shifted to the red side of the spectrum. The interpretation that this is due to a "stretching" of the universe is only true when applied to FLRW. Note: you cannot think of this in terms of a Doppler effect.
This suggests that it is equally plausible that light from a far off galaxy that was emitted long ago was emitted (at the time of its emission) with a spectrum that was already shifted to the red side of the spectrum. And, if so, this would have different ramification regarding the expansion of the universe as well as the rate at which temperature changes as the universe ages.
The point is that the same observation can support more than one model and one must be careful not to think of any of our current models (General Relativity, Standard Model, etc) as being absolute. They are merely models that - to this point - have served us well - but they should not be viewed as absolute.
Therefore, what do we know about the temperature of the universe in the future? Only what our current models suggest, which may be very incomplete.
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In a dark-energy-only far future, the Universe will asymptote to the de Sitter temperature ($T_{ds}$), the minimum temperature possible in the Universe, which is NOT absolute zero (absolute zero is un-physical, the 3rd law of thermodynamics was actually quantified in 2016). In natural units:
$T_{ds}=\frac{1}{2πl} \approx 2.4\times10^{-30}[K] $
Where $l$ is the radius of the future cosmic event horizon (approx. $16.1Gly$).
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