  Recent works have related the bulk first law of black hole mechanics to the
first law of entanglement in a dual CFT. These are first order relations, and
receive corrections for finite changes. In particular, the latter is naively
expected to be accurate only for small changes in the quantum state. But when
Newton's constant is small relative to the AdS scale, the former holds to good
approximation even for classical perturbations that contain many quanta. This
suggests that -- for appropriate states -- corrections to the first law of
entanglement are suppressed by powers of $N$ in CFTs whose correlators satisfy
't Hooft large-$N$ power counting. We take first steps toward verifying that
this is so by studying the large-$N$ structure of the entropy of spatial
regions for a class of CFT states motivated by those created from the vacuum by
acting with real-time single-trace sources. We show that $1/N$ counting matches
bulk predictions, though we require the effect of the source on the modular
hamiltonian to be non-singular. The magnitude of our sources is $\epsilon N$
with $\epsilon$ fixed-but-small as $N\rightarrow \infty$. Our results also
provide a perturbative derivation -- without relying on the replica trick -- of
the subleading Faulkner-Lewkowycz-Maldacena correction to the Ryu-Takayagi and
Hubeny-Rangamani-Takayanagi conjectures at all orders in $1/N$.
