I want to estimate the the graviton temperature at $T=1$MeV assuming that the gravitons decoupled at $T \sim m_\text{pl}$, the Planck mass.
My reasoning was the following. We know that due to conservation of entropy that $g_S(T)T^3a^3$ is constant, where $g_S(T)$ is the effective number of degrees of freedom in entropy. From the famous graph on the effective number of relativistic degrees of freedom in cosmology, we have that before the graviton decouples, we have $g_S(T \sim m_\text{pl} ) = 106.75 +2$ and at $T = 1 \text{MeV}$ $g_S(T = 1 \text{MeV}) = 10.75$.
As a result we obtain $$108.75T^3_G a^3_G = 10.75 T^3 a^3.$$
How do I proceed from here?
I'm tempted to use $T \propto a^{-1}$, but this does seem to hold always I believe.
