  We exploit exact inequalities that refer to the entropy of a distribution to
derive a simple variational principle at finite temperature for trial density
matrices of Gutzwiller and Jastrow type. We use the result to extend at finite
temperature the Gutzwiller approximation, which we apply to study a two-orbital
model that we believe captures some essential features of V$_2$O$_3$. We indeed
find that the phase diagram of the model bears many similarities to that of
real vanadium sesquioxide. In addition, we show that in a Bethe lattice, where
the finite temperature Gutzwiller approximation provides a rigorous upper bound
of the actual free energy, the results compare well with the exact phase
diagram obtained by the dynamical mean field theory.
