Universal temperature corrections to the free energy for the gravitational field

The temperature correction to the free energy of the gravitational field is considered which does not depend on the Planck energy physics. The leading correction may be interpreted in terms of the temperature dependent effective gravitational constant $G_{\mathrm{eff}}$. The temperature correction to $G_{\mathrm{eff}}^{-1}$ appears to be valid for all temperatures $T\ll E_{\mathrm{Planck}}$. It is universal since it is determined only by the number of fermionic and bosonic fields with masses $m\ll T$, does not contain the Planck energy scale $E_{\mathrm{Planck}}$ which determines the gravitational constant at $T=0$, and does not depend on whether or not the gravitational field obeys the Einstein equations. That is why this universal modification of the free energy for gravitational field can be used to study thermodynamics of quantum systems in condensed matter (such as quantum liquids superfluid $^3$He and $^4$He), where the effective gravity emerging for fermionic and/or bosonic quasiparticles in the low-energy corner is quite different from the Einstein gravity.