Erickalle: For heat capacities we have the law of Dulong & Petit, that says that the heat capacity of solids is about 25 joule per mole per kelvin. This is about the same for metals and nonmetals. From this you can see that the heat capacity of the electron gas is much smaller than what a classical theory would predict.
So instead, you have to look at it as degenerate gas of fermions, with electron energies that follow the FermiDirac distribution. The heat capacity is then of order kT/Ef smaller than the classical prediction. The Fermi energy Ef of metals is of the order of 3 eV. At room temperature kT is about 25 meV. So that is a ratio of 10^2.
