Mass and Euclidean action of $m+n$ dimensional black holes
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In this subsection we show another surprising result that the mass of these black holes also vanishes. To do this, we employ the Euclidean approach to black hole thermodynamics. The Euclidean action $I_E$ of the black holes includes two parts, the bulk and boundary parts, $$I_E=I+B\, ,$$ where $I$ is the bulk action, while $B$ denotes the boundary term. The bulk part is given by $$\label{reducedaction}
I=-\left(r_0^{D-m}\Omega_{D-m}\right)
\left\{\sum_{t=0}^{s}A_t\int\sqrt{g}d^{m}x\bar{L}_{t}\right\}\, ,$$ where $\bar{L}_{t}$ is given by (\[Lbart\]). With the metric (\[m-2metric\]), we get $$\label{bulkaction} I=-\left(r_0^{D-m}\Omega_{D-m}V_{m-2}\right)
\left\{\sum_{t=0}^{s}A_t\frac{(m-2)!}{(m-2t)!}\int d\tau  dr
\left(r^m F^t\right)''\right\}\, ,$$ where $V_{m-2}$ is the volume of the $\Sigma_{m-2}$ with unit radius. In general, the boundary term $B$ is a little bit complicated. For simplicity, we only consider here the case that the highest derivative term is the Gauss-Bonnet term, i.e., we deal with the cases with $m=3$ and $m=4$. In that case, the boundary term is given by [@Myers:1987yn; @Davis:2002gnGravanis:2002wy] $$B=-2\int_{\partial X}d^{D-1}v\sqrt{h}\left[K+2c_2
\left(J-2E_{\mu\nu}K^{\mu\nu}\right)\right]\, ,$$ where we have set $c_1=1$ and $h$ is the induced metric on a timelike boundary $\partial X$. The tensor $K^{\mu\nu}$ is the extrinsic curvature of the boundary, and $K$ is its trace. We denote by $J$ the 