We present geometric derivations of the Smarr formula for static AdS black
holes and an expanded first law that includes variations in the cosmological
constant. These two results are further related by a scaling argument based on
Euler's theorem. The key new ingredient in the constructions is a two-form
potential for the static Killing field. Surface integrals of the Killing
potential determine the coefficient of the variation of the cosmological
constant in the first law. This coefficient is proportional to a finite,
effective volume for the region outside the AdS black hole horizon, which can
also be interpreted as minus the volume excluded from a spatial slice by the
black hole horizon. This effective volume also contributes to the Smarr
formula. Since the cosmological constant is naturally thought of as a pressure,
the new term in the first law has the form of effective volume times change in
pressure that arises in the variation of the enthalpy in classical
thermodynamics. This and related arguments suggest that the mass of an AdS
black hole should be interpreted as the enthalpy of the spacetime.Comment: 21 pages; v2 references adde
