  Recently, a novel formula for computing entropy in theories coupled to
semi-classical gravity has been devised. Using this so-called island formula
the entropy of semi-classical black holes follows a Page curve. Here, we study
the relation between this novel entropy and semi-classical entropy in the
context of doubly-holographic models. Double holography allows for two
different $d$-dimensional descriptions of a black hole coupled to a
non-gravitational bath, both of which allow a holographic computation of von
Neumann entropy in bath subregions. We argue that the correct homology
constraint for Ryu-Takayanagi surfaces depends on which of those
$d$-dimensional perspectives is taken. As a consequence the von Neumann
entropies of a fixed subregion in both descriptions can disagree. We discuss
how the von Neumann entropies in both descriptions are related to the entropy
computed by the island formula and coarse grained entropy. Moreover, we argue
that the way operators transform between the two descriptions depends on their
complexity. A simple toy model is introduced to demonstrate that a sufficiently
complicated map between two descriptions of the system can give rise to an
island formula and wormholes. Lastly, we speculate about the relation between
double-holography and black hole complementarity.
