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 non-metals. 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 Fermi-Dirac 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.