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Perturbative Metric of Charged Black Holes in Quadratic Gravity

We consider perturbative solutions to the classical field equations coming from a quadratic gravitational lagrangian in four dimensions. We study the charged, spherically symmetric black hole and explicitly give corrections up to third order (in the coupling constant $β$ multiplying the $R_{μν}R^{μν}$ term) to the Reissner--Nordström hole metric. We consider the thermodynamics of such black holes, in particular, we compute explicitly its temperature and entropy--area relation which deviates from the usual $S=A/4$ expression.