No evidence for violation of the second law in extended black hole thermodynamics

Recently, a number of papers have claimed that the horizon area-and thus the entropy-of near extremal black holes in anti-de Sitter spacetimes can be reduced by dropping particles into them. In this paper, we point out that this is a consequence of an underlying incorrect assumption that the energy of an infalling particle changes the internal energy of the black hole by the same amount, whereas actually it is the mass or enthalpy of the black hole that increases by the energy of the particle absorbed.