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 V2O3. 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 dynamical mean-field theory.