where $\tilde{q}_{ij}$ is the induced metric of the cross section of the horizon in the $m$-dimensional part.

By the similar procedure to get (\[emm\]) and (\[enn\]), we have $$\begin{aligned}
\mathfrak{S}&=&\sum_{t=0}^{w}\sum_{k=t+1}^{p}\Bigg{\{}{k-1 \choose
t}4\pi k c_k
\frac{(D-m)!}{(D-m-2(k-1-t))!}\left(\frac{\bar{k}}{r_0^2}\right)^{k-t-1}\nonumber\\
&~~~~~~~\times& \int d^{D-m}yr_0^{D-m} \sqrt{\gamma}\int
d^{m-2}z\mathcal{L}_{t}(\tilde{q})\Bigg{\}}\, ,\end{aligned}$$ where $w=[(m-2)/2]$. Define $\ell=t+1$, we have $$\begin{aligned}
\label{ent}
\mathfrak{S}&=&\sum_{\ell=1}^{w+1}\Bigg{\{}4\pi
\ell\Bigg{[}\sum_{k=\ell}^{p}{k \choose \ell} c_k
\frac{(D-m)!}{(D-m-2(k-\ell))!}\left(\frac{\bar{k}}{r_0^2}\right)^{k-\ell}
\Bigg{]}\nonumber\\
&~~~~~~~\times& \int d^{D-m}yr_0^{D-m} \sqrt{\gamma}\int
d^{m-2}z\mathcal{L}_{\ell-1}(\tilde{q})\Bigg{\}}\, .\end{aligned}$$ Thus the entropy can be expressed as $$\begin{aligned}
\label{3eq22}
\mathfrak{S}=\sum_{\ell=1}^{w+1}\Bigg{\{}4\pi \ell
A_{\ell} \int d^{D-m}yr_0^{D-m} \sqrt{\gamma}\int
d^{m-2}z\mathcal{L}_{\ell-1}(\tilde{q})\Bigg{\}}\, ,\end{aligned}$$ where $A_{\ell}$ is defined in Eq. (\[lineq\]), from which we have $$\label{Alequation} A_{\ell}=0\, ,\quad \ell=1,\cdots ,[(m-1)/2]\, .$$ Thus we find from the entropy (\[3eq22\]) that when $m$ is odd, $\mathfrak{S}=0$, a vanishing entropy!

When $m$ is even, the entropy is $$\label{entropy1} \mathfrak{S}=2\pi m
\left(r_0^{D-m}\Omega_{D-m}\right)c_{m/2}\chi(\Sigma_{m-2})\, .$$ where $\Omega_{D-m}$ is the volume of $\mathcal{H}^n$, and $\chi(\Sigma_{m-2})$ is the integration of Euler characteristic on the $(m-2)$-dimensional cross section of horizon surface, i.e., $$\chi(\Sigma_{m-2})=\int_{\Sigma_{m-2}}
d^{m-2}z\mathcal{L}_{(m/2-1)}(\tilde{q})\, .$$ For the cross section of horizon surface $\Sigma_{m-2}$, which is a constant curvature space, $\chi(\Sigma_{m-2})$ is constant, while in the case of $\Sigma_{m-2}$ being a closed manifold, $\Sigma_{m-2}$ need not be a constant curvature space, and in that case, $\chi(\Sigma_{m-2})$ is the Euler number of $\Sigma_{m-2}$ up to a 