Jane, you had to mean quantum statistical mechanics. Thermodynamics only emerges in the thermodynamic limit of statistical mechanics, which requires a large number of degrees of freedom, and when you have a large number of degrees of freedom, the relevant quantities automatically behave classically, too. In this sense, thermodynamics is always classical (non-quantum).
Quantum statistical physics is easily promoted to special relativity. But one must understand that a thermal ensemble is given by
$$\rho\approx \exp(-\beta H)$$
where $H$ is the energy, the generator of translations in time. You may extend it in a relativistically invariant way,
$$\rho\approx \exp(-\beta H - \vec\beta_p \cdot \vec p)$$
where $\vec\beta_p$ are spatial components of inverse temperature. However, the density matrix above is equivalent - by a Lorentz transformation - to the previous one. Only the norm of the 4-vector $(\beta,\beta_p)$ has an invariant physical meaning, and you may always go to the frame where the spatial components vanish. In this frame, you get an ordinary thermal ensemble.
(Of course, another question is whether one knows how to calculate the thermal properties of objects in a particular relativistic theory. That involves new mathematical challenges. But the definition of the objects - e.g. the thermal density matrix - is totally unaffected by having a relativistic theory.)
So the formula for the thermal state is actually identical for relativistic quantum statistical mechanics - and obviously, it's also identical in the classical limit of it, the relativistic classical statistical mechanics. The 4-vector of the inverse temperature had to be time-like, otherwise the density matrix would be divergent (the momentum-dominated "energy" wouldn't be bounded from below).
In general relativity, you may define the temperature (and its preferred reference frame) locally, and locally, the physics reduces to that of special relativity which was explained above. However, there's no real ensemble with a fixed global temperature globally, except for geometries that are static - that have a time-like Killing vector field. That's because to define the thermal ensemble, you need a periodic Euclidean time (the energy generating time translations enters the exponential in the density matrix), and if the original solution nontrivially depended on time, you couldn't make the time periodic in the imaginary direction.
There are many fancy thermal effects in curved spaces - such as particle production; Unruh effect; and Hawking radiation. The most exciting portion is black hole thermodynamics. However, to get there, one must first understand why the extension of statistical mechanics to the special relativistic context isn't really conceptually new. It may be useful if you asked a more advanced question about the references when you figure out what is actually out there so that your question becomes somewhat more well-defined.