  We explore the complexity equals volume proposal for planar black holes in
anti-de Sitter (AdS) spacetime in 2+1 dimensions, with an end of the world
(ETW) brane behind the horizon. We allow for the possibility of intrinsic
gravitational dynamics in the form of Jackiw-Teitelboim (JT) gravity to be
localized on the brane. We compute the asymptotic rate of change of volume
complexity analytically and obtain the full time dependence using numerical
techniques. We find that the inclusion of JT gravity on the brane leads to
interesting effects on time dependence of holographic complexity. We identify
the region in parameter space (the brane location and the JT coupling) for
which the rate of change of complexity violates the Lloyd bound. In an
equivalent description of the model in terms of an asymptotically AdS wormhole,
we connect the violation of the Lloyd bound to the violation of a suitable
energy condition in the bulk that we introduce. We also compare the Lloyd bound
constraints to previously derived constraints on the bulk parameters in this
model that are based on bounds on entanglement growth in the dual CFT state.
