  We investigate the complexity=volume proposal in the case of Janus AdS$_3$
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.
