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In the Complexity-Action (CA) proposal there are two different methods to regularize the gravitational on-shell action, which are equivalent in the framework of AdS/CFT. In this paper, we want to study the equivalence of them in the context of AdS/BCFT. To do so, we consider a thermal BCFT state in two dimensions which is dual to a two-sided BTZ black hole excised by a dynamical ETW brane. We show that if one includes the timelike counterterms inspired by holographic renormalization as well as the Gibbons-Hawking-York term on the timelike boundary of the WDW patch, which exists in one of the regularizations, the coefficients of the UV divergent terms of action-complexity in the two methods become equal to each other.
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