The usual explanation given for why electrons do not contribute much to the specific heat of a material (for the purposes of the question consider a conductor) is that the fermi energy of conductors is pretty large (for copper, it's 7 eV) compared to the thermal heat produced (0.026 eV) in the conductor at about 300 K, so that only electrons in the band within 0.026 eV of the fermi level can thermally participate.
I have an obvious question here - if we are actually supplying heat to such a conductor, what would happen if the energy we're supplying is large compared to the 7 eV of the Fermi level? When I think of gases, it makes sense that the specific heat should be independent of the heat supplied, but in the case of conductors (considering the electronic contribution) I think it's more like:
greater the heat supplied ---> more electrons participate thermally ---> more electrons absorb that heat and thus increase the heat capacity.
Of course, I am making a foolish mistake here, but I can't tell where exactly. Also, I don't know anything about solid state physics, since I'm only in high school.