  We use the gravitational path integral (GPI) to compute the fluctuations of
the Hawking radiation entropy around the Page curve, in a two-dimensional model
introduced by Penington \emph{et al}. Before the Page time, we find that
$\delta S = e^{-S}/\sqrt{2}$, where $S$ is the black hole entropy. This result
agrees with the Haar-averaged entropy fluctuations of a bipartite system, which
we also compute at leading order. After the Page time, we find that $\delta S
\sim e^{-S}$, up to a prefactor that depends logarithmically on the width of
the microcanonical energy window. This is not symmetric under exchange of
subsystem sizes and so does not agree with the Haar average for a subsystem of
fixed Hilbert space dimension. The discrepancy can be attributed to the fact
that the black hole Hilbert space dimension is not fixed by the state
preparation: even in a microcanonical ensemble with a top-hat smearing
function, the GPI yields an additive fluctuation in the number of black hole
states. This result, and the fact that the Page curve computed by the GPI is
smooth, all point towards an ensemble interpretation of the GPI.
