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We investigate the complexity=volume proposal in the case of Janus AdS$_3$geometries, both at zero and finite temperature. The leading contributioncoming from the Janus interface is a logarithmic divergence, whose coefficientis a function of the dilaton excursion. In the presence of the defect,complexity is no longer topological and becomes temperature-dependent. We alsostudy the time evolution of the extremal volume for the time-dependent JanusBTZ black hole. This background is not dual to an interface but to a pair ofentangled CFTs with different values of the couplings. At late times, when theequilibrium is restored, the couplings of the CFTs do not influence thecomplexity rate. On the contrary, the complexity rate for theout-of-equilibrium system is always smaller compared to the pure BTZ black holebackground.

Volume complexity for Janus $\mathrm{AdS}_3$ geometries