= Solution
Only states within an <energy> shell of width roughly $k_BT$ around $E_F$ appreciably change occupation at low <temperature>. There are roughly $g(E_F)k_BT$ such states and each excitation costs <energy> of order $k_BT$. Thus $\Delta U\propto g(E_F)(k_BT)^2$, giving \b[$C_{el}\propto g(E_F)k_B^2T$]. This is the <thermal-shell estimate of electronic heat capacity>; filled states far below $E_F$ are blocked by the <Pauli exclusion principle>. A smooth DOS and fixed particle number give the more precise Sommerfeld result $C_{el}=(\pi^2/3)g(E_F)k_B^2T$ to leading order.
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