as $F=F_b+\Delta F$ with the leading order correction $\Delta F$. We arrive at $$\Delta F\sum_{i=0}^{s} i ~\widehat{\alpha}_iF_b^{i-1}= \Delta
F\widehat{\alpha}=\frac{M}{r^{m-1}}   \, .$$ Here the constant $\widehat{\alpha}$ is given by $$\widehat{\alpha}=\sum_{i=0}^{s} i ~\widehat{\alpha}_iF_b^{i-1}\, .$$ For solutions in some branch, $\widehat{\alpha}$ may be negative. However, we only consider the cases with positive $\widehat{\alpha}$ here. So we find the expansion of metric around the background as $$g_{tt}-g_{(b)tt}= -f+f_{b}=r^2\Delta F \approx
\frac{M}{\widehat{\alpha}~ r^{m-3}}=\frac{16\pi
G_{m}\mathfrak{M}}{(m-2)V_{m-2}r^{m-3}}\, .$$ Thus we find that the mass of the black hole $\mathfrak{M}$ has a relation to $M$ as $$\label{massm} \frac{M}{\widehat{\alpha}}=\frac{16\pi
G_{m}\mathfrak{M}}{(m-2)V_{m-2}}\, ,$$ where $V_{m-2}$ is the volume of $(m-2)$-dimensional cross section of horizon surface. The mass can be expressed in terms of horizon radius $r_+$ and $Q$ as $$\mathfrak{M}=\frac{(m-2)V_{m-2}}{16\pi
G_{m}\widehat{\alpha}}\left(-\frac{Q}{r_{+}}+
r_{+}^{m-1}\sum_{i=0}^{s}\widehat{\alpha}_{i}\left(\frac{\delta}{r_{+}^2}\right)^i\right)\,
,$$ and its variation with respect to the horizon radius is $$\left(\frac{\partial \mathfrak{M}}{\partial
r_{+}}\right)_{Q}=\frac{(m-2)V_{m-2}}{16\pi
G_{m}\widehat{\alpha}}r_{+}^{m-4}\left(\sum_{i=0}^{s}(m-2i-1)
\widehat{\alpha}_{i}\delta\left(\frac{\delta}{r_{+}^2}\right)^{i-1}+\frac{Q}{r_{+}^{m-2}}\right)\,
.$$ Since we are dealing with black holes in higher derivative gravity theory, the well-known area formula for black hole entropy breaks down. Let us try to obtain the black hole entropy by integrating the first law of the black hole thermodynamics $$\label{integration}
\mathfrak{S}=\int \mathfrak{T}^{-1} d
\mathfrak{M} =\int^{r_+} \mathfrak{T}^{-1}\left(\frac{\partial
\mathfrak{M}}{\partial r_{+}}\right)_{Q}dr_{+}\, .$$

(i). When $m$ is even, $s$ takes the value $m/2$. The integral 