The difference between the 170,000 year "diffusion time" and the ~10 million year "Kelvin-Helmoltz time" (which is normally defined in terms of gravitational energy, but this is related by factors like 2 that I'm not concerned with here) has to do with what energy you are tracking. If you are only tracking radiant energy, you get the former, but if you are also tracking the heat in the gas, you get the latter. So yes, I think that's what you mean by the heat content of the two.