I'd agree with the observations that the literature is a bit of a mess, and it's hard to tell from it if a consensus view even exists.
My personal point of view is that entropy is a simpler concept than temperature, entropy being basically a scalar density. You can regard is as the log of the number of states using the POV of statistical thermodynamics.
Be warned that my views of thermo are basically based on reading one paper that I found persuasive, rather than any in-depth research. Said paper is
http://arxiv.org/abs/physics/0505004.
This paper itself admits that there is/was a bit of controversy on the issue, and mentions some of the competing ideas that I never really looked into deeply.
To oversimplify a lot, we start with the point of view that entropy, being the log of the number of states, is a world scalar. More precisely, entropy per unit volume is a scalar density. Going on from this starting point, the traditional idea is that entropy, work (energy) and temperature are related by the equation ##\Delta S = \Delta Q / T##. But in special relativity, energy is one component of a 4-vector, so our first step in expressing the above law covariaintly replaces ##\Delta Q## with a 4 vector, which represents the exchange of energy-momentum between systems rather than the exchange of a scalar energy. Since we want to map a change in energy-momentum to a change in a scalar value, we need another 4-vector which replaces inverse temperature in the equation. Thus inverse temperature becomes not a scalar, but a 4-vector.
There is at least one issue that remains to be fixed, though. This is the fact that energy-momentum is only a 4-vector for an isolated system, and we often wish to treat non-isolated systems in thermodynamics. We can still treat the entropy per unit volume as a scalar density, but the energy/momentum per unit volume is best treated via the stress-energy tensor. We still keep the concept of inverse temperature as a 4-vector, and we represent the energy/momentum per unit volume via the stress energy tensor.
On a more practical note, all the textbook treatmens I have just choose some particular frame, and make a point of doing the thermodynamics in a non-covariant manner, rather than worry about the issue of how to do it covariantly. I'd say this is probably the easiest approach to communicate.
