= Solution
<Pauli exclusion principle> prevents most filled electronic states from changing occupation at $T\ll T_F$. Only a shell of width of order $k_BT$ around the <Fermi energy> participates. Since the <density of states> near that <energy> is of order $n/E_F$ per <volume> and <energy>, the fraction of active <electrons> is of order $k_BT/E_F=T/T_F$. Their typical change of <energy> is of order $k_BT$, so the <thermal energy> increment per <volume> scales as
$$
\Delta u_e\sim n\frac{T}{T_F}k_BT.
$$
Differentiating with respect to <temperature> gives the <electronic heat capacity>
$$
\boxed{C_e=A\,nk_B\frac{T}{T_F}}.
$$
The simple shell argument fixes the linear law, not its coefficient. A <Sommerfeld expansion> using the spin-summed free-electron <density of states> $g(E_F)=3n/(2E_F)$ gives $C_e=(\pi^2/3)g(E_F)k_B^2T$, hence $A=\pi^2/2$, consistent with the supplied value. The result excludes strong band-structure or interaction corrections and requires a degenerate <electron> population.
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