use the systematic method to regulate the gravitational action of asymptotically adS solutions which is through the use of the counterterm method. It was shown that the counterterm approach become quite reasonable when applied to AdS/CFT, as the boundary counterterm has a natural interpretation as conventional field theory counterterm that show up in the dual CFT [@Mal].

The counterterm action is a functional of the boundary curvature invariants and do not affect on the symmetries and field equations of the bulk $\mathcal{M}$ $${I_{\mathrm{ct}}}=\int_{\partial \mathcal{M}}{{d^{n}}x\sqrt{-h}L(l,\widehat{R%
},\nabla \widehat{R},...)}.  \label{Ict1}$$

In a general manner, the counterterm in Lovelock gravity is a scalar constructed from the curvature invariants of the boundary as in the case of Einstein gravity [@Kraus; @DehNew]. Although one can use the procedure of Ref. [@DehNew] to compute the counterterm action for arbitrary horizon topology, for the sake of brevity and simplification, we deal with the spacetime with zero curvature boundary ($\widehat{R}_{abcd}(\gamma )=0$). In this case all the counterterm containing the curvature invariants of the boundary are zero (see [@TOLNLED; @DehShah2] for more details) and the counterterm reduces to $$\mathcal{I}_{\mathrm{ct}}=\frac{1}{8\pi }\int_{\partial
\mathcal{M}}d^{n}x\sqrt{-\gamma }\left(
\frac{n-1}{l_{\mathrm{eff}}}\right) ,\   \label{Ict}$$where $l_{\mathrm{eff}}$ is given by

$$l_{\mathrm{eff}}=\frac{15\sqrt{\alpha \left[ 1-\left( 1-\frac{3\alpha }{l^{2}%
}\right) ^{1/3}\right] }}{9\left( 1+\frac{\alpha }{l^{2}}\right) -\left[
