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The first law for entanglement entropy in CFT in an odd-dimensionalasymptotically AdS black hole is studied by using the AdS/CFT duality. Theentropy of CFT considered here is due to the entanglement between twosubsystems separated by the horizon of the AdS black hole, which itself isrealized as the conformal boundary of a black droplet in even-dimensionalglobal AdS bulk spacetime. In $(2+1)$-dimensional CFT, the first law is shownto be always satisfied by analyzing a class of metric perturbations of theexact solution of a $4$-dimensional black droplet. In $(4+1)$-dimensions, thefirst law for CFT is shown to hold under the Neumann boundary condition at acertain bulk hypersurface anchored to the conformal boundary of the boundaryAdS black hole. From the boundary view point, this Neumann condition yieldsthere being no energy flux across the boundary of the boundary AdS black hole.Furthermore, the asymptotic geometry of a $6$-dimensional small AdS blackdroplet is constructed as the gravity dual of our $(4+1)$-dimensional CFT,which exhibits a negative energy near the spatial infinity, as expected fromvacuum polarization.

The first law of entanglement entropy in AdS black hole backgrounds