I have two theories. [edit: theories why anyone would come up with such an idea in the first place, that is]
First, E = energy, and temperature is energy. So if temperature increases, doesn't that increase E? And if E = mc², doesn't that mean that either mass or the speed of light would need to speed up in order to keep up with it?
Second, although false, a lot of people are trained to believe that time stands still at 0 K. In that case, light could never escape 0 K, and as temperatures approach 0 K light would slog to a halt. If that was the case, the logical conclusion would be that speed of light would increase as temperatures rise.
Or maybe something completely different - I just thought it was a fun question to try to reverse engineer. :)