  We develop an extension of the Gutzwiller approximation to finite
temperatures based on the Dirac-Frenkel variational principle. Our method does
not rely on any entropy inequality, and is substantially more accurate than the
approaches proposed in previous works. We apply our theory to the single-band
Hubbard model at different fillings, and show that our results compare
quantitatively well with dynamical mean field theory in the metallic phase. We
discuss potential applications of our technique within the framework of first
principle calculations.
