Why is scale factor inversely proprotional to temperature?

There is a nice clear discussion of this in the paper Cooling in the Universe by Sohrab Rahvar. We are not supposed to give link only answers, but I'm not sure what is to be gained by just duplicating the working in Rahvar's paper. Basically you define temperature by the associated random thermal velocities, then write down the geodesic equation for these in an FLRW spacetime. This is realtively straightforward as only a few of the Christoffel symbols are non-zero, and the result falls out that the velocities scale as $a^{-1}$. That means the temperature scales as $a^{-1}$.