7$.

Thermodynamic properties of black hole solutions
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Naive consideration: $m$-dimensional black holes
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The black hole spacetime has a direct product form $\mathcal{M}^m
\times \mathcal{H}^{n}$, where $\mathcal{H}^{n}$ is a negative constant curvature space with a constant radius $r_0$. From the point of view of usual Kaluza-Klein dimensional reduction, the thermodynamics for the whole spacetime is equivalent to that for $m$-dimensional black hole with redefined gravitational constant $G_m=G_{m+n}/{\rm Vol}(n)$. Here $G_m$ and $G_{m+n}$ are gravitational constants in $m$ dimensions and $(m+n)$ dimensions, respectively, while ${\rm Vol}(n)$ is the volume of the constant curvature space $\mathcal{H}^{n}$. In this subsection, we will discuss the black hole thermodynamics from the point of view of $m$ dimensions.

Assume that black hole has a horizon at $r_+$, which is the largest positive root of $f(r)=0$. The horizon radius then must satisfy $$F(r_{+})=\frac{\delta}{r_{+}^2}\, , \quad \mathrm{or} \quad
r_{+}^2F(r_{+})= \delta\, .$$ The Hawking temperature of the black hole can be easily calculated by Euclidean method. To avoid conical singularity at the horizon, the period of Euclidean time should be $ \beta = 4\pi/f'(r_+) $, and the Hawking temperature is just the inverse of the period. This way we get the temperature of the black hole $$\label{temperature}
\mathfrak{T}=\frac{1}{\beta}=\frac{1}{4\pi}f^{'}(r_{+})=-\frac{1}{4\pi}
