  We present a comprehensive study of the thermodynamic properties of the
three-dimensional fermionic Hubbard model, with application to cold fermionic
atoms subject to an optical lattice and a trapping potential. Our study is
focused on the temperature range of current experimental interest. We employ
two theoretical methods - dynamical mean-field theory and high-temperature
series - and perform comparative benchmarks to delimitate their respective
range of validity. Special attention is devoted to understand the implications
that thermodynamic properties of this system have on cooling. Considering the
distribution function of local occupancies in the inhomogeneous lattice, we
show that, under adiabatic evolution, the variation of any observable (e.g.,
temperature) can be conveniently disentangled into two distinct contributions.
The first contribution is due to the redistribution of atoms in the trap during
the evolution, while the second one comes from the intrinsic change of the
observable. Finally, we provide a simplified picture of the cooling procedure
recently proposed in J.-S. Bernier et al., Phys. Rev. A 79, 061601 (2009) by
applying this method to an idealized model.
