  In the framework of the static patch approach to de Sitter holography
introduced in [arXiv:2109.14104], the growth of holographic complexity has a
hyperfast behaviour, which leads to a divergence in a finite time. This is very
different from the AdS spacetime, where instead the complexity rate
asymptotically reaches a constant value. We study holographic volume complexity
in a class of asymptotically AdS geometries which include de Sitter bubbles in
their interior. With the exception of the static bubble case, the complexity
obtained from the volume of the smooth extremal surfaces which are anchored
just to the AdS boundary has a similar behaviour to the AdS case, because it
asymptotically grows linearly with time. The static bubble configuration has a
zero complexity rate and corresponds to a discontinuous behaviour, which
resembles a first order phase transition. If instead we consider extremal
surfaces which are anchored at both the AdS boundary and the de Sitter
stretched horizon, we find that complexity growth is hyperfast, as in the de
Sitter case.
