  Leaky boundary conditions in asymptotically AdS spacetimes are relevant to
discuss black hole evaporation and the evolution of the Page curve via the
island formula. We explore the consequences of leaky boundary conditions on the
one-loop partition function of gravity. We focus on JT gravity minimally
coupled to a scalar field whose normalizable and non-normalizable modes are
both turned on, allowing for leakiness through the AdS boundary. Classically,
this yields a flux-balance law relating the scalar news to the time derivative
of the mass. Semi-classically, we argue that the usual diffeomorphism-invariant
measure is ill-defined, suggesting that the area-non-preserving diffeomorphisms
are broken at one loop. We calculate the associated anomaly and its implication
on the gravitational Gauss law. Finally, we generalize our arguments to higher
dimensions and dS.
