Thermodynamical aspects of running vacuum models
  The thermal history of a large class of running vacuum models in which the
effective cosmological term is described by a truncated power series of the
Hubble rate, whose dominant term is $\Lambda (H) \propto H^{n+2}$, is discussed
in detail. Specifically, by assuming that the ultra-relativistic particles
produced by the vacuum decay emerge into space-time in such a way that its
energy density $\rho_r \propto T^{4}$, the temperature evolution law and the
increasing entropy function are analytically calculated. For the whole class of
vacuum models explored here we findthat the primeval value of the comoving
radiation entropy density (associated to effectively massless particles) starts
from zero and evolves extremely fast until reaching a maximum near the end of
the vacuum decay phase, where it saturates. The late time conservation of the
radiation entropy during the adiabatic FRW phase also guarantees that the whole
class of running vacuum models predicts thesame correct value of the present
day entropy, $S_{0} \sim 10^{87-88}$ (in natural units), independently of the
initial conditions. In addition, by assuming Gibbons-Hawking temperature as an
initial condition, we find that the ratio between the late time and primordial
vacuum energy densities is in agreement with naive estimates from quantum field
theory, namely, $\rho_{\Lambda 0}/\rho_{\Lambda I} \sim10^{-123}$. Such results
are independent on the power $n$ and suggests that the observed Universe may
evolve smoothly between two extreme, unstable, nonsingular de Sitter phases.
