  From the black hole thermodynamics point of view, we show that the entropy
function $\mathbf{f}$ and the free energy $F$ are related via
$\mathbf{f}=e_{I}q_{I}+\Omega_Hq_I{A_{\phi}^I}'-\frac{\partial F}{\partial r}
|_{r_{H}}$. Assuming the entropy function is known for extremal black holes, we
propose an approach to calculate the entropy of non-extremal cases by slightly
moving the extremal black hole geometry from extremality. The entropy of
non-extremal $D1D5$- and $D1D5p$-branes in the presence of higher derivative
corrections are computed as concrete examples. An attempt has also been made to
explain why the entropy function method can calculate the corrected entropy
without knowing the exact form of black hole solution in higher derivative
gravity theories.
