  We use the Einstein-Hilbert gravitational path integral to investigate
gravitational entanglement at leading order $O(1/G)$. We argue that
semiclassical states prepared by a Euclidean path integral have the property
that projecting them onto a subspace in which the Ryu-Takayanagi or
Hubeny-Rangamani-Takayanagi surface has definite area gives a state with a flat
entanglement spectrum at this order in gravitational perturbation theory. This
means that the reduced density matrix can be approximated as proportional to
the identity to the extent that its Renyi entropies $S_n$ are independent of
$n$ at this order. The $n$-dependence of $S_n$ in more general states then
arises from sums over the RT/HRT-area, which are generally dominated by
different values of this area for each $n$. This provides a simple picture of
gravitational entanglement, bolsters the connection between holographic systems
and tensor network models, clarifies the bulk interpretation of algebraic
centers which arise in the quantum error-correcting description of holography,
and strengthens the connection between bulk and boundary modular Hamiltonians
described by Jafferis, Lewkowycz, Maldacena, and Suh.
