eternal two-sided black holes. Thus we have right and left null infinities in the Penrose diagram of these spacetimes. In order to control divergent terms we need to restrict the WDW patch by using some cutoffs. Therefore the boundary of space-time on which the on-shell action must be computed consists of null surfaces besides timelike ones and their joint points. A complete computation requires that we accompany boundary terms to the bulk action. Hence we use the following generic action: $$\begin{aligned}
\nonumber
I =& \frac{1}{16 \pi G_{N}} \int_{\mathcal{M}} d^{d+1} x \sqrt{-g}~\mathcal{R}
\\\nonumber
 &+ \frac{1}{8 \pi G_{N}} \int_{\mathcal{B}} d x^d  \sqrt{|h|} ~K + \frac{1}{8 \pi G_{N}} \int _{\Sigma}d^{d-1}x \sqrt{\sigma} ~\eta
 \\\label{action}
 &+\frac{1}{8 \pi G_{N}} \int_{\mathcal{B'}} d x^d \sqrt{\gamma} ~\kappa + \frac{1}{8 \pi G_{N}} \int _{\Sigma}d^{d-1}x \sqrt{\sigma} ~a\end{aligned}$$ The first term is related to the volume of the WDW patch and is vanishing in the flat scenario. The Vanishing of the bulk term in the on-shell action is the most important technical difference between the AdS case and the flat case.

The second line of action belongs to the non-null boundaries. The first term is known as the Gibbons-Hawking-York (GHY) term in the spacelike and timelike sector of the boundary. The GHY 