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We express the Noether charge entropy density of a black brane inanti-deSitter space in terms of local operators in the anti-deSitter spacebulk. We find that Wald's expression for the Noether charge entropy needs to bemodified away from the horizon by an additional term that vanishes on thehorizon. We then determine the field theory dual of the Noether charge entropyfor theories that asymptote to Einstein theory. We do so by calculating thevalue of the entropy density at the anti-deSitter space boundary and applyingthe standard rules of the AdS/CFT correspondence. We interpret the variation ofthe entropy density operator from the horizon to the boundary as due to therenormalization of the effective gravitational couplings as they flow from theultra-violet to the infra-red. We discuss the cases of Einstein-Hilbert theoryand f(R) theories in detail and make general comments about more complicatedcases.

The Noether charge entropy in anti-deSitter space and its field theory dual