  The first law for entanglement entropy in CFT in an odd-dimensional
asymptotically AdS black hole is studied by using the AdS/CFT duality. The
entropy of CFT considered here is due to the entanglement between two
subsystems separated by the horizon of the AdS black hole, which itself is
realized as the conformal boundary of a black droplet in even-dimensional
global AdS bulk spacetime. In $(2+1)$-dimensional CFT, the first law is shown
to be always satisfied by analyzing a class of metric perturbations of the
exact solution of a $4$-dimensional black droplet. In $(4+1)$-dimensions, the
first law for CFT is shown to hold under the Neumann boundary condition at a
certain bulk hypersurface anchored to the conformal boundary of the boundary
AdS black hole. From the boundary view point, this Neumann condition yields
there being no energy flux across the boundary of the boundary AdS black hole.
Furthermore, the asymptotic geometry of a $6$-dimensional small AdS black
droplet is constructed as the gravity dual of our $(4+1)$-dimensional CFT,
which exhibits a negative energy near the spatial infinity, as expected from
vacuum polarization.
