How to Calculate the On-Shell Action for an AdS Schwarzschild Black Hole?

[Jesus]

Homework Statement

I want to calculate the on shell action for the case of a AdS Schwarzschild black hole.

Homework Equations

Following the case of the flat Schwarzschild black hole I tried to add a counter-term of the type of a Gibbons-Hawking boundary term for flat Minkowski spacetime, however this doesn't seem to lead to a finite answer when I take the limit r to infinity.

Is there a method to find out what is the right counter-term that I need to add?

 

[member 553083]

The Attempt at a SolutionI have tried to use the same method as for the flat Schwarzschild black hole, i.e. adding a counter-term of the type of a Gibbons-Hawking boundary term for flat Minkowski spacetime. However this doesn't seem to lead to a finite answer when I take the limit r to infinity.