Entropy, energy and temperature-length inequality for Friedmann
  universes
  In this paper we continue the study of the physical consequences of our
modified black hole entropy formula in expanding spacetimes. In particular, we
apply the new formula to apparent horizons of Friedmann expanding universes
with zero, negative and positive spatial curvature. As a first result, we found
that, apart from the static Einstein solution, the only Friedmann spacetimes
with constant (zero) internal energy are the ones with zero spatial curvature.
This happens because, in the computation of the internal energy $U$, the
contribution due to the non-vanishing Hubble flow must been added to the usual
Misner-Sharp energy giving, for zero curvature spacetimes, a zero value for
$U$. This fact does not hold when curvature is present. After analyzing the
free energy $F$, we obtain the correct result that $F$ is stationary only for
physical systems in isothermal equilibrium, i.e. a de Sitter expanding
universe. This result permits us to trace back a physically reasonable
hypothesis concerning the origin of the early and late times de Sitter phase of
our universe. Finally, we deduce an interesting temperature-length inequality
similar to the time-energy uncertainty of ordinary quantum mechanics but with
temperature instead of time coordinate. Remarkably, this relation is
independent on the gravitational constant $G$ and can thus be explored also in
non gravitational contexts.
