Electrons in the top triangle are being excited to the bottom triangle due to temperature increase. Number of excited electrons $\approx\frac{1}{4}g(\varepsilon_{\rm F})k_{\rm B}T=n_{\rm exc}$. Total extra energy $E(T)-E(0)=n_{\rm exc}k_{\rm B}T=\frac{1}{2}g(\varepsilon_{\rm F})k_{\rm B}^2T^2$.
Electrons in the top triangle are being excited to the bottom triangle due to temperature increase. Number of excited electrons $\approx\frac{1}{4}g(\varepsilon_{\rm F})k_{\rm B}T=n_{\rm exc}$. Total extra energy $E(T)-E(0)=n_{\rm exc}k_{\rm B}T=\frac{1}{4}g(\varepsilon_{\rm F})k_{\rm B}^2T^2$.