The entropy expressions (\[3eq16\]) and (\[3eq17\]) look quite similar to the entropy formula of static, spherically symmetric black holes in Lovelock gravity [@Cai4], except for the logarithmic term in (\[3eq16\]). The appearance of the logarithmic term is strange, although such a term appears in the entropy expressions of black holes in Horava-Lifshitz gravity theory [@CCO], while the latter is not a full diffeomorphism invariant theory. For a diffeomorphism invariant gravity theory, Wald showed that black hole entropy is a Noether charge [@Wald:1993nt]; further a well-known entropy formula was developed [@Wald:1993nt; @Iyer:1994ys]. By Wald’s entropy formula, black hole entropy is given by some integral on the black hole horizon. Therefore black hole entropy must be a function of horizon geometry and a logarithmic term will never appear in Wald’s entropy formula. This may cause suspicion whether the results given above are valid or not.

Let us notice that the above way to obtain black hole solution in $m$ dimensions is quite different from the usual Kaluza-Klein dimensional reduction. In the usual Kaluza-Klein theory, with the assumption of direct product of the manifold $\mathcal{M}^m\times \mathcal{H}^n$, one gets reduced action by integrating the total action over the extra space $\mathcal{H}^n$. Certainly here too, 