Black hole thermodynamics: No inconsistency via the inclusion of the missing P-V terms

Understand

The early literature on black hole thermodynamics ignored the $P$-$V$ term associated with the existence of a fundamental physical constant in the black hole solution.

• The inclusion of this constant in the first law becomes inconsistent with the Smarr relation.
• Once the missing $P$-$V$ term introduced is, it becomes customary to introduce it only in problems where there is a negative cosmological constant.
• This practice is inherited from cosmological approaches which consider the quantity $-\Lambda/8\pi$ as the constant pressure due to a cosmological fluid.