What is the ratio of specific heat in the universe as a whole? Is there any measure of the ratio of specific heat in the universe as a whole?
 A: The specific heat of the universe as a whole is the subject of this study titled Cosmographic study of the universe’s specific heat: A landscape for Cosmology? by Orlando Luongo and Hernando Quevedo published in April 2013.

A  method is proposed for constructing the specific heat for the universe by following standard
definitions of classical thermodynamics, in a spatially flat homogeneous and isotropic spacetime. We
use cosmography to represent the specific heat in terms of measurable quantities, and show that a
negative specific heat at constant volume and a zero specific heat at constant pressure are compatible
with observational data.
We derive the most general cosmological model which is compatible with
the values obtained for the specific heat of the universe, and show that it overcomes the fine-tuning
and the coincidence problems of the ΛCDM model.

The equations formulated for specific heats are $$C_P=\frac{V_0}{4\pi G}\frac{H^2(j-1)}{T'(1+z^4)} \\ C_V=\frac{V_0}{8\pi G}\frac{H^2(2q-1)}{T'(1+z^4)}$$
These reveal the specific heats as functions of redshift. While for zero redshift, the equations reduce to  $$C_{P0}=\frac{V_0}{4\pi G}\frac{H^2(j-1)}{T'} \\ C_{V0}=\frac{V_0}{8\pi G}\frac{H^2(2q-1)}{T'}$$
and the numerical results can be obtained,  which are $$C_{P0}=-0.030^{+0.748}_{-0.793}\times 10^{52}JK^{-1} \\
C_{V0}=-1.587^{+0.156}_{-0.151}\times 10^{52}JK^{-1}$$ the specific heats at constant $P$ and $V$ respectively.
As can be seen, these are huge values for specific heat as one would expect for the universe as a whole. The ratio of specific heats (as per your question)  is then $$\gamma=\frac{C_{P0}}{C_{V0}}\approx 0.0189$$ (without the error bounds).
