Rotational energy is angular momentum times angular velocity. So in case where angular velocity is reduced by 1/gamma, because of time dilation, then rotational energy is reduced by 1/gamma. So the thermal energy in the rotational degree of freedom goes as 1/gamma.
Force on the 'wall' of an Einstein light clock is reduced by 1/gamma. So the amount of work the photon gas inside the clock can do goes as 1/gamma. So thermal energy of photon gas goes as 1/gamma.
A somewhat related thought:
A black body that moves very fast emits photons at rate that goes as 1/gamma, it emits at slow rate. But it still absorbs photons normally, so that the black body tends to absorb thermal energy, which is a definition of low temperature.