When higher derivative corrections are considered, the entropy does not scale any more like the horizon area. The black-hole entropy at equilibrium can still be defined as the quantity that obeys the first law of black-hole mechanics, where the mass, angular momenta and other extensive quantities are defined with all higher-derivative corrections included. More precisely, the entropy is first defined for non-extremal black holes by integrating the first law, and using properties of non-extremal black holes, such as the existence of a bifurcation surface [262, 176]. The resulting entropy formula is unique and given by

where is the binormal to the horizon, i.e., the volume element of the normal bundle to . One can define it simply as , where is the generator of the horizon and is an outgoing null normal to the horizon defined by and . Since the Lagrangian is diffeomorphism invariant (possibly up to a boundary term), it can be expressed in terms of the metric, the matter fields and their covariant derivatives, and the Riemann tensor and its derivatives. This operator acts on the Lagrangian while treating the Riemann tensor as if it were an independent field. It is defined as a covariant Euler–Lagrange derivative as Moreover, the entropy formula is conserved away from the bifurcation surface along the future horizon as a consequence of the zeroth law of black-hole mechanics [178]. Therefore, one can take the extremal limit of the entropy formula evaluated on the future horizon in order to define entropy at extremality. Quite remarkably, the Iyer–Wald entropy (52) can also be reproduced [20] using Cardy’s formula as we will detail below.In five-dimensional Einstein gravity coupled to matter, the entropy of extremal black holes can be expressed as

where and have been defined in (36) and .From the attractor mechanism for four-dimensional extremal spinning black holes [17], the entropy at extremality can be expressed as an extremum of the functional

where is the Lagrangian. The entropy then only depends on the angular momentum and the conserved charges , and depend in a discontinuous fashion on the scalar moduli [240]. The result holds for any Lagrangian in the class (1), including higher-derivative corrections, and the result can be generalized straightforwardly to five dimensions.When quantum effects are taken into account, the entropy formula also gets modified in a non-universal way, which depends on the matter present in quantum loops. In Einstein gravity, the main correction to the area law is a logarithmic correction term. The logarithmic corrections to the entropy of extremal rotating black holes can be obtained using the quantum entropy function formalism [241].

Living Rev. Relativity 15, (2012), 11
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