discussions by two examples of $m=3$ and 4.

### The case of $m=3$, $p=2$, $D\ge 5$

In this case, the metric $\tilde{h}$ is very simple, which is just the metric of the constant curvature space $\mathcal{H}^n$ plus the metric of a circle, i.e., $$\tilde{h}= dz^2 + r_{0}^2 \gamma_{ab}dy^a dy^b\, ,$$ and Wald’s entropy (\[entropywald\]) becomes $$\label{m34entropy} \mathfrak{S}=4\pi  \int d^{D-2}x+8\pi c_2\int
d^{D-2}xR(\tilde{h})\, .$$ Here we have set $c_1$ to unity as before, and $R(\tilde{h})$ is the scalar curvature of the metric $\tilde{h}$. It is easy to find $$R(\tilde{h})=(D-3)(D-4)\left(\frac{\bar{k}}{r_0^2}\right)\, ,$$ thus we have $$\mathfrak{S}=4\pi
(r_0^{D-3}\Omega_{D-3})V_1\left[1+2c_2(D-3)(D-4)\left(\frac{\bar{k}}{r_0^2}\right)\right]
\, ,$$ where $V_1$ is the volume of the circle. Considering $\bar{k}=-1$, and the explicit relation among $c_2$, $r_0$ and $D$ in Eq. (\[c0c2r0D\]), the entropy identically vanishes.

Actually, for $m=3$, we have $w=0$, so $\ell$ in Eq. (\[3eq22\]) can take the value $1$ only. Since $A_{1}=0$ by Eq. (\[Alequation\]), we reach a vanishing entropy from (\[3eq22\]), as we have just shown above.

### The case of $m=4$, $p=2$, $D\ge 6$

In this case, the metric $\tilde{h}$ consists of the metric of the constant curvature space $\mathcal{H}^n$ and the metric of a 2-dimensional constant curvature space $\Sigma_2$, i.e., $$\tilde{h}= \tilde{q}_{ij}dz^idz^j + r_{0}^2 \gamma_{ab}dy^a dy^b\, ,$$ and Wald’s 