  We study the entanglement phase structure of a holographic boundary conformal
field theory (BCFT) in a two-dimensional black hole background. The bulk dual
is the AdS$_3$ black string geometry with a Karch-Randall brane. We compute the
subregion entanglement entropy of various two-sided bipartitions to elucidate
the phase space where a Page curve exists in this setup. We do fully analytical
computations on both the gravity side and the field theory side and demonstrate
that the results precisely match. We discuss the entanglement phase structure
describing where a Page curve exists in this geometry in the context of these
analytical results. This is a useful model to study entanglement entropy for
quantum field theory on a curved background.
