The first law and weak cosmic censorship for de Sitter black holes

We apply Iyer–Wald’s covariant phase space formalism to asymptotically de Sitter spacetimes and establish the thermodynamic first law, expressed in terms of the Abbott–Deser mass. Similar to Iyer–Wald’s first law for asymptotically flat black holes, our first law applies to general asymptotically de Sitter perturbations around a Reissner–Nordström–de Sitter black hole, without imposing any symmetry for the perturbations. We explicitly derive the first law up to the second order in perturbation. Additionally, we apply this first law to a thought experiment involving the overcharging of a Reissner–Nordström–de Sitter black hole by injecting energy and charge sources. We find that asymptotically de Sitter black holes cannot be overcharged, provided that the null energy condition holds, thereby reinforcing the weak cosmic censorship conjecture.