In his undergraduate text "Spacetime Physics", Wheeler points out that there is always a fourth component to momentum and energy interactions, because the internal motion of the objects involved will change over time for reasons different from their macroscopic motion. The deformation of impact creates heat, for instance. Outside of energy known to be produced by some process within the object, is this the only contributor?
If that makes sense, consider this thought-experiment:
If we imagine a totally ordered solid, no vibration, at absolute zero, etc., then every particle in it contributes to its total mass exactly the rest mass of the particle itself.
So let time pass. The objects entropy will increase. That means that the particles are now moving relative to the frame of reference of the solid's center of mass. So it should increase very slightly in mass. Each moving particle will see a slight increase due to the effects of its motion relativistically.
Therefore, we should be able to measure the entropy within the mass by detecting the difference between its exhibited mass and the total rest mass of its constituents.
Does this hold water? Could this effect reliably define temperature, and entropy, in general, relativistically? I doubt we can discern the actual rest masses with such accuracy that this is useful. But is it theoretically true?