We consider the Complexity=Action (CA) proposal, in Einstein gravity, and investigate
counterterms which are able to remove all the UV divergences of holographic complexity.
To study the UV divergences, one can apply two different methods of regularizations. It
has been proved [22, 23] that the coefficients of the UV divergences are not equal in both
regularizations. We show that if one considers the GibbonsHawkingYork term as well as
new types of counterterms inspired from holographic renormalization on timelike boundaries
of the WDW patch, the coefficients on both sides match very well. Next, we introduce
new counterterms on the null boundaries of the WDW patch for four and five dimensional
asymptotically AdS spacetimes which are able to remove all the UV divergences of the on
shell action. Moreover, they are covariant and do not change the equations of motion. At
the end, we calculate them for an AdSSchwarzschild black hole.
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