  In the action-complexity 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
for a pure BTZ black hole microstate. The microstate is obtained from a
two-sided BTZ black hole truncated by a dynamical timelike ETW brane. Moreover,
it is dual to a finite energy pure state in a two-dimensional CFT. 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. Furthermore, we compare the finite terms of
action-complexity in both regularizations, and show that when the UV cutoff
surface is close enough to the asymptotic boundary of the bulk spacetime,
action-complexities in both regularizations become exactly equal to each other.
