constant factor. For example, when $m=4$, $D=6$, we have $$\mathfrak{S}= 64\pi^2\cdot \delta\cdot c_2 \cdot(r_0^2 \Omega_2)\, .$$ Clearly we see that when $\Sigma_2$ is a Ricci flat space, i.e. $\delta =0$, the constant entropy vanishes. Here the constant means that it is independent of the horizon radius $r_+$ and charge $Q$.

Now we argue that the constant entropy is meaningless for black hole thermodynamics and should be dropped. One simple reason is that when the cross section of horizon surface $\sigma_{m-2}$ is a negative constant curvature space, $\chi$ is negative, giving a negative entropy which does not make sense in thermodynamics. Another reason is provided by the following example. Consider a four-dimensional Schwarzschild black hole solution. In this case, the Euler density is the Gauss-Bonnet term. If one considers the contribution of the Gauss-Bonnet term to the black hole entropy, besides the usual area entropy, one has an additional constant $\mathfrak{S}= 64\pi^2 c_2$ from (\[entropy1\]), where $c_2$ is the Gauss-Bonnet coefficient. Such a constant term remains even when the black hole horizon goes to zero. Both of these clearly indicate that the constant entropy from the horizon topology should be dropped when black hole thermodynamics is concerned.

Let us illustrate these 