= Solution
For $T\ll T_F$, only <electrons> within an energy window of order $kT$ around the <Fermi energy> can change occupation appreciably, because of the <Pauli exclusion principle>. The active fraction is of order $kT/E_F=T/T_F$. Giving each of these <electrons> energy of order $kT$ estimates an excitation energy $U-U_0\sim N_ekT^2/T_F$, and therefore
$$
\boxed{C_V\sim\frac{2N_ekT}{T_F}}.
$$
Per mole of conduction <electrons>, $N_ek=R$, which is the quoted $2RT/T_F$. Per mole of metal atoms with $z$ model conduction <electrons> each, the <conduction-electron count in the molar electronic heat capacity> instead gives $2zRT/T_F$. The numerical factor is only approximate: the <Sommerfeld expansion> gives $C_V=(\pi^2/2)N_ekT/T_F$ for a three-dimensional free-electron gas.
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