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The holographic complexity of formation for the AdS3 2-sided Randall-Sundrum model and the AdS3/BCFT2 models is logarithmically divergent according to the volume conjecture, while it is finite using the action proposal. One might be tempted to conclude that the UV divergences of the volume and action conjectures are always different for defects and boundaries in two-dimensional conformal field theories. We show that this is not the case. In fact, in Janus AdS3 we find that both volume and action proposals provide the same kind of logarithmic divergences.

We investigate the complexity=volume proposal in the case of Janus AdS3 geometries, both at zero and finite temperature. The leading contribution coming from the Janus interface is a logarithmic divergence, whose coefficient is a function of the dilaton excursion. In the presence of the defect, complexity is no longer topological and becomes temperature-dependent. We also study the time evolution of the extremal volume for the time-dependent Janus BTZ black hole. This background is not dual to an interface but to a pair of entangled CFTs with different values of the couplings. At late times, when the equilibrium is restored, the couplings of the CFTs do not influence the complexity rate. On the contrary, the complexity rate for the out-of-equilibrium system is always smaller compared to the pure BTZ black hole background.