  The volume inside a Ryu-Takayanagi surface has been conjectured to be related
to the complexity of subregions of the boundary field theory. Here, we study
the behaviour of this volume analytically, when the entangling surface has a
strip geometry. We perform systematic expansions in the low and high
temperature regimes for AdS-Schwarzschild and RN-AdS black holes. In the latter
regime, we point out spurious divergences that might occur due to the
limitations of a near horizon expansion. A similar analysis is performed for
extremal black holes, and at large charge, we find that there might be some new
features of the volume as compared to the area. Finally, we numerically study a
four dimensional RN-AdS black hole in global AdS, the entangling surface being
a sphere. We find that the holographic complexity captures essentially the same
information as the entanglement entropy, as far as phase transitions are
concerned.
