Does relativity objectively define entropy? 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?
 A: So, unfortunately, if a box contains a certain amount of kinetic/potential energy stored within it, that energy appears as a mass to the outside world.
This is actually the basis for using the so-called "mass defect" (change in mass) in radioactive reactions for detecting the energy released: One over-large Uranium nucleus becomes some smaller parts with lower total mass. Did a nucleon disappear? Not really, if you ignore neutrinos flying off undetectably -- but the nucleus just has stored potential energy in the strong nuclear force which was appearing as the "extra mass" to the rest of the world which was converted to kinetic energy in the products of the reaction.
Consequently, when you shift energy from potential to kinetic in the box, the observed mass from the outside doesn't change.
Instead, the reason that the object at absolute 0 has an "increasing entropy" is purely due to thermal energy transfer with its surroundings, which will appear as energy -- hence mass -- going into the box. Measuring the mass very carefully will, as John Rennie says in a comment to your question, reveal a change in the total internal energy, but not anything deep or fundamental about the entropy contained within it: it could have gotten that mass because its temperature increased, but it could also have gotten that mass in the form of positronium (electron-positron pairs) which are a tenth of a microsecond away from exploding into gamma rays. You don't really know just by looking at the mass of the box which has happened.
