Extremal black holes and the first law of thermodynamics

We study the low-temperature expansion of the first law of thermodynamics for near-extremal black holes. We show that for extremal black holes with nonvanishing entropy, the leading-order contribution yields an expression for their extremal entropy that is in agreement with the entropy-function result. When their entropy vanishes due to the vanishing of a one-cycle on the horizon, such a leading contribution is always compatible with the first law satisfied by a Bañados-Teitelboim-Zanelli black hole. The universality of these results follows from universal facts about extremal black holes. Our results are consistent with both the presence of local AdS₂ and AdS₃ near-horizon throats for extremal black holes and with the suggested quantum microscopic descriptions (AdS₂/CFT₁, Kerr/CFT, and extremal vanishing horizon/CFT).