ChrisVer said:
I think adiabatic implies the conservation of energy?
As has already been discussed in this thread, you have to be careful with the concept of "energy" when dealing with the universe as a whole, for two reasons: first, it isn't closed (according to our best current model), so its "total energy" is infinite; second, it's expanding, so it's not stationary, which means the whole idea of "conservation of energy" is problematic.
"Adiabatic", strictly speaking, means no heat transfer (or matter transfer) between the system under study and anything else. The "system under study" in this case is the "cosmological fluid", the homogeneous, isotropic, continuous system of stress-energy that fills the universe in the standard cosmological model. The adiabatic assumption means there is no significant heat transfer between this system and any other matter or energy in the universe. However, that does not necessarily mean the energy of that system is constant (see below).
In the reference you gave, since the key focus of the section you pointed out is entropy, the main component of the "fluid" under study is the CMBR--a photon bath which is taken to be in thermal equilibrium, and which contains the vast majority of the entropy in the universe. However, that does not mean it contains the vast majority of the energy in the universe; and, in fact, it doesn't--it contains a couple of orders of magnitude less energy than ordinary matter (which is itself only a few percent of the total energy according to our best current model--the rest is dark matter and dark energy).
Also, since the universe is expanding, even if we restrict attention to the photon "fluid", and treat its expansion as adiabatic, the total energy of the fluid is not constant, because the expansion redshifts the photons. Before the "surface of last scattering", when the CMBR was emitted, the photons were in thermal equilibrium with the matter (which was a plasma then), so heat was being transferred from the matter to the photons to counteract the redshift of the photons as the universe expanded. But when the matter "recombined" (a strange term since it had never been "combined" before) into atoms and stopped being a plasma, the photons became decoupled from the matter, and ever since, they have been losing energy relative to the matter because of the redshift. This "lost" energy has not been transferred anywhere; it is a manifestation of the fact that, as noted above, global "conservation of energy" is a problematic concept in a non-stationary spacetime.
As far as I can tell, the reference you gave does not use global energy conservation to justify the adiabatic approximation; it only uses continuity of mass-energy and equation 3.2.58, which relates pressure and temperature in equilibrium.