= Solution
At $T\ll T_F$, Pauli exclusion restricts thermally changed occupations to an <energy> interval of order $k_BT$ about the <Fermi energy>. The affected fraction is of order $T/T_F$; their typical excitation <energy> is of order $k_BT$, so the thermal <energy> correction is proportional to $T^2$ and the <heat capacity> is linear in $T$.
More precisely, with $g(E_F)$ the total <density> of one-particle states per unit <energy>, including <spin>,
$$
\boxed{C_V=\frac{\pi^2}{3}g(E_F)k_B^2T+O(T^3).}
$$
For a three-dimensional <ideal gas> with quadratic dispersion, $g(E_F)=3N/(2E_F)$, hence
$$
\boxed{C_V=\frac{\pi^2}{2}Nk_B\frac{T}{T_F},\qquad T_F=E_F/k_B.}
$$
The fixed-particle-number <Sommerfeld expansion> gives $U=3NE_F/5+(\pi^2/4)N(k_BT)^2/E_F+\cdots$. The <temperature> dependence of the <chemical potential> must be included when obtaining this expression; holding it artificially fixed changes the intermediate <energy> correction.
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