What an excellent question! Let me first give a little introduction for other readers.
The First Law is just the conservation of energy. Applying it in general to problems in General Relativity is tricky (and too hard for me).
Anyway, here's the expansion-cooling story. Take some thermal light in a region of space, with the temperature of the thermal radiation being T. Now let the space stretch out so that all the dimensions are twice as big, leaving the volume 8 times as big, The stretching of space doubles the wavelength of each photon, reducing its energy by a factor of 2. Since there's 8 times the volume, the energy density is down by a factor of 16. That's exactly what happens when you cut the temperature of thermal radiation by a factor of 2. In a simple expansion of the space filled with thermal radiation, the temperature falls inversely proportional to the linear dimensions.
OK, now the tricky part. It's not the factor of 8 that's hard- that obviously comes from energy conservation itself, with the radiation being more spread out. It's the factor of 2 that's the issue. How can each photon lose energy if energy is conserved? Here I can only repeat words I've heard. If you include the gravitational interaction energy density, there's a negative term in the energy density which just cancels the other positive terms. So you start with zero energy and end up with zero energy. However, not all papers treat this issue this way, and some claim that the energy conservation idea is not useful in this context.
(published on 12/17/2010)