On black hole thermodynamics and the entropy function formalism
  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. 
Xian-Hui Ge and Fu-Wen Shu