Noether charge entropy in anti-de Sitter space and its field theory dual

We express the Noether charge entropy density of a black brane in anti-de Sitter space in terms of local operators in the anti-de Sitter space bulk. We find that Wald's expression for the Noether charge entropy needs to be modified away from the horizon by an additional term that vanishes on the horizon. We then determine the field theory dual of the Noether charge entropy for theories that asymptote to Einstein theory. We do so by calculating the value of the entropy density at the anti-de Sitter space boundary and applying the standard rules of the AdS/CFT correspondence. We interpret the variation of the entropy density operator from the horizon to the boundary as due to the renormalization of the effective gravitational couplings as they flow from the ultraviolet to the infrared. We discuss the cases of Einstein-Hilbert theory and f(R) theories in detail and make general comments about more complicated cases.