we have to consider the whole $(m+n)$-dimensional theory. This is very different from the usual Kaluza-Klein theory in which the effective action is just the reduced action. As a result, we cannot simply use Wald’s entropy formula to get the entropy of the $m$-dimensional black hole. On the other hand, Lovelock theory is diffeomorphism invariant and Wald’s entropy formula is applicable in the whole $(m+n)$ dimensions. In the following subsections, we discuss thermodynamics of the black holes by Euclidean action and Wald’s entropy formula in $(m+n)$ dimensions, and find quite different and surprising results.

Entropy of $(m+n)$-dimensional black holes
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From the viewpoint of the whole $D(=m+n)$ dimensions, to study thermodynamics of these black holes is straightforward. The temperature of the black holes is the same as the one in Eq. (\[temperature\]) because it is determined by the horizon geometry only. In Lovelock gravity, Wald’s entropy formula can be expressed as [@Jacobson:1993xs] $$\label{entropywald} \mathfrak{S}=\sum_{k=1}^{p}4\pi k c_k \int
d^{D-2}x\mathcal{L}_{k-1}(\tilde{h})\, ,$$ where $\mathcal{L}_{k-1}(\tilde{h})$ has the same form as (\[Lk\]) except that metric is replaced by $\tilde{h}$, which is the induced metric on the $(D-2)$-dimensional cross section of the horizon. The induced metric $\tilde h$ is $$\tilde{h}= \tilde{q}_{ij}dz^idz^j + r_{0}^2 \gamma_{ab}dy^a dy^b\, ,$$ 