 Solution

Only states within an energy shell of width roughly $k_{B}T$ around $E_F$ appreciably change occupation at low temperature. There are roughly $g(E_F)k_{B}T$ such states and each excitation costs energy of order $k_{B}T$. Thus $\Delta U\propto g(E_F)(k_{B}T{)}^2$, giving $C_{el}\propto g(E_F)k_B^{2}T$. 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 $C_{el}=({\pi }^2/3)g(E_F)k_B^{2}T$ to leading order.