= Solution
For a degenerate <Fermi gas>, the <Pauli exclusion principle> restricts thermally changed occupations to an energy shell of width $k_BT$ around the <Fermi energy>. The number of active <Electrons> is of order $g(E_F)k_BT\sim Nk_BT/E_F$; each carries an excitation energy of order $k_BT$. Thus the thermal energy grows as $N(k_BT)^2/E_F$ and the <electronic heat capacity> is proportional to $T$.
More precisely, the fixed-particle-number <Sommerfeld expansion> gives
$$
U(T)-U(0)=\frac{\pi^2}{6}g(E_F)(k_BT)^2+O(T^4),\qquad
C_e=\frac{\pi^2}{3}g(E_F)k_B^2T+O(T^3).
$$
Here $g$ is the total <spin-summed density of states> for the sample. For a three-dimensional parabolic band, $g(E_F)=3N/(2E_F)$, yielding
$$
\boxed{C_e=A\frac{Nk_B^2T}{E_F},\qquad A=\frac{\pi^2}{2},\qquad k_BT\ll E_F.}
$$
The linear law concerns the electronic contribution; the low-temperature lattice <heat capacity> usually adds a $T^3$ term.
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