I finally located a reference that was sleeping in the back of my mind:
"Relativity, Thermodynamics and Cosmology", Richard C. Tolman, Oxford,
printings 1934 -1962
has several chapters on thermodynamic implications of both SR and GR.
For SR he arrives at T = T_0*Sqrt(1-[u/c]^2} as the temperature of an object
with speed u (Eq 69.19, pg 158).
I have not plowed through this material - it was noticed years ago and has
been a repressed lurker in my subconscious ever since.
If anyone knows of a review or commentary on this text, I would be grateful.