The thermodynamics of a gravitating vacuum
  In the present days of modern cosmology it is assumed that the main ingredient to cosmic energy presently is vacuum energy with an energy density $\epsilon_\mathrm{vac}$ that is constant over the cosmic evolution. In this paper here we show, however, that this assumption of constant vacuum energy density is unphysical, since it conflicts with the requirements of cosmic thermodynamics. We start from the total vacuum energy including the negatively valued gravitational binding energy and show that cosmic thermodynamics then requires that the cosmic vacuum energy density can only vary with cosmic scale $R=R(t)$ according to $\epsilon _\mathrm{vac}\sim R^{-\nu }$ with only two values of $\nu$ being allowed, namely $\nu _\mathrm{1}=2$ and $\nu _\mathrm{2}=5/2$. We then discuss these two remaining solutions and find, when requiring a universe with a constant total energy, that the only allowed power index is $\nu _\mathrm{1}=2$. We discuss the consequences of this scaling of $\epsilon _\mathrm{vac}$ and show the results for a cosmic scale evolution of a quasi-empty universe like the one that we are presently faced by. 
M. Heyl, H.J. Fahr, M. Siewert
physics.gen-ph gr-qc