  The basic idea of a microscopic understanding of Thermodynamics is to derive
its main features from a microscopic probability distribution. In such a vein,
we investigate the thermal statistics of quasi-probabilities's semi-classical
analogs in phase space for the important case of quadratic Hamiltonians,
focusing attention in the three more important instances, i.e., those of
Wigner, $P$-, and Husimi distributions. Introduction of an effective
temperature permits one to obtain a unified thermodynamic description that
encompasses and unifies the three different quasi-probability distributions.
This unified description turns out to be classical.
