  In AdS/CFT, how is the bulk first law realized in the boundary CFT? Recently,
Faulkner et al. showed that in certain holographic contexts, the bulk first law
has a precise microscopic interpretation as a first law of entanglement entropy
in the boundary theory. However, the bulk can also satisfy a first law when the
boundary density matrix is pure, i.e. in the absence of entanglement with other
degrees of freedom. In this note we argue that the bulk first law should
generally be understood in terms of a particular coarse-graining of the
boundary theory. We use geons, or single-exterior black holes, as a testing
ground for this idea. Our main result is that for a class of small
perturbations to these spacetimes the Wald entropy agrees to first order with
the one-point entropy, a coarse-grained entropy recently proposed by Kelly and
Wall. This result also extends the regime over which the one-point entropy is
known to be equal to the causal holographic information of Hubeny and
Rangamani.
