This paper presents geometric derivations of the Smarr formula and an extended first law for static AdS black holes, including variations in the cosmological constant. The key ingredient is a two-form potential for the static Killing field, which determines the coefficient of the variation of the cosmological constant in the first law. This coefficient is proportional to an effective volume outside the black hole horizon, which can also be interpreted as minus the volume excluded by the black hole horizon. The effective volume contributes to the Smarr formula and is interpreted as the enthalpy of the spacetime, analogous to the classical thermodynamic concept of enthalpy. The new term in the first law has the form of effective volume times change in pressure, similar to the variation of enthalpy in thermodynamics. The results are related by a scaling argument based on Euler's theorem. The paper also discusses the implications of including the cosmological constant as a thermodynamic variable, and its potential applications in the AdS/CFT correspondence. The results are generalized to Lovelock gravity theories, and the paper concludes with a discussion of the Killing potential near a static horizon.This paper presents geometric derivations of the Smarr formula and an extended first law for static AdS black holes, including variations in the cosmological constant. The key ingredient is a two-form potential for the static Killing field, which determines the coefficient of the variation of the cosmological constant in the first law. This coefficient is proportional to an effective volume outside the black hole horizon, which can also be interpreted as minus the volume excluded by the black hole horizon. The effective volume contributes to the Smarr formula and is interpreted as the enthalpy of the spacetime, analogous to the classical thermodynamic concept of enthalpy. The new term in the first law has the form of effective volume times change in pressure, similar to the variation of enthalpy in thermodynamics. The results are related by a scaling argument based on Euler's theorem. The paper also discusses the implications of including the cosmological constant as a thermodynamic variable, and its potential applications in the AdS/CFT correspondence. The results are generalized to Lovelock gravity theories, and the paper concludes with a discussion of the Killing potential near a static horizon.