A bstract We investigate the holographic timelike entanglement entropy (HTEE) and timelike subregion complexity of a thermal CFT d deformed by a relevant scalar operator ϕ 0 , dual to a hairy black hole in AdS d +1 . We employ the prescription of merging spacelike and timelike surfaces at the interior, constructing an extremal surface homologous to a boundary timelike subsystem with a time interval ∆ t . Consequently, this deformation breaks the invariance of the imaginary component of HTEE observed in pure AdS 3 and BTZ geometry, introducing a nontrivial dependence on ∆ t . At small ∆ t , we derive analytical expressions that are in agreement with numerical results, and observe partial consistency with analytic continuation to temporal or spacelike entanglement entropy at the level of the near-boundary expansion. However, analytic continuation of CFT temporal entanglement entropy fails to reproduce the HTEE calculations under boundary deformation, even in d = 2. Furthermore, we extend the numerical calculations to higher dimensions ( d = 3). In addition, we study holographic timelike subregion complexity within the complexity=volume conjecture and find that it remains real-valued, providing a complementary geometric probe of the black hole interior. In particular, for the BTZ black hole, we analytically show that the UV-finite term of the subregion complexity receives its entire contribution from the interior region alone.
Prihadi et al. (Wed,) studied this question.