gives $$\begin{aligned}
\label{3eq16}
\mathfrak{S}&=&\int \mathfrak{T}^{-1} d \mathfrak{M}
=\int^{r_+} \mathfrak{T}^{-1}\left(\frac{\partial
\mathfrak{M}}{\partial
r_{+}}\right)_{Q}dr_{+}\nonumber\\
&=&\frac{V_{m-2}}{4
G_{m}\widehat{\alpha}}\left[\sum_{i=1}^{s-1}\frac{m-2}{m-2i}
i~\widehat{\alpha}_{i}\delta ^{i-1}r_+^{m-2i} +s(s-1)
\widehat{\alpha}_s~\delta^{s-1}~\mathrm{ln}
\left(r_+^2\right)\right] +\mathfrak{S}_0\, .\end{aligned}$$ The last term $\mathfrak{S}_0$ is an integration constant. Note that here a logarithmic term appears, which comes from the fact that $s$ can take the value $m/2$.

(ii). When $m$ is odd, $s$ is $(m-1)/2$. In this case, the integral gives $$\begin{aligned}
\label{3eq17}
 \mathfrak{S}&=&\int \mathfrak{T}^{-1} d \mathfrak{M}
=\int^{r_+} \mathfrak{T}^{-1}\left(\frac{\partial
\mathfrak{M}}{\partial
r_{+}}\right)_{Q}dr_{+}\nonumber\\
&=&\frac{V_{m-2}}{4
G_{m}\widehat{\alpha}}\left[\sum_{i=1}^{s}\frac{m-2}{m-2i}
i~\widehat{\alpha}_{i}\delta ^{i-1}r_+^{m-2i}\right]+ \mathfrak{S}_0
\, .\end{aligned}$$ Here $\mathfrak{S}_0$ is also an integration constant. Note that here the integration constant $\mathfrak{S}_0$ should be set to zero because when the black hole horizon shrinks to zero, the entropy of the black hole should vanish [@Cai1]. However, the integration constant cannot be fixed by the same argument in the case of even $m$, due to the existence of the logarithmic term in the black hole entropy. In addition, let us notice that when the black hole horizon is a Ricci flat surface, namely, $\delta=0$, not only does the logarithmic term disappear in (\[3eq16\]), but also both (\[3eq16\]) and (\[3eq17\]) give an entropy proportional to horizon area. This is also a general feature of black hole entropy in Lovelock gravity [@Cai4; @Cai1].

