The paper presents geometric derivations of the Smarr formula and an extended first law for static AdS black holes, incorporating 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 \(\Lambda\) in the first law. This coefficient is proportional to an effective volume outside the AdS black hole horizon, which can also be interpreted as minus the volume excluded by the black hole horizon. This effective volume contributes to the Smarr formula. The new term in the first law, \(\Theta \delta \Lambda / 8\pi G\), where \(\Theta\) is the effective volume, resembles the term \(V \delta P\) in the variation of enthalpy in classical thermodynamics. This suggests that the mass of an AdS black hole should be interpreted as the enthalpy of the spacetime. The paper also discusses the physical interpretation of \(\Theta\) and its role in the first law, and provides a geometric derivation of the Smarr formula and the first law for AdS black holes using the Komar integral relation.The paper presents geometric derivations of the Smarr formula and an extended first law for static AdS black holes, incorporating 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 \(\Lambda\) in the first law. This coefficient is proportional to an effective volume outside the AdS black hole horizon, which can also be interpreted as minus the volume excluded by the black hole horizon. This effective volume contributes to the Smarr formula. The new term in the first law, \(\Theta \delta \Lambda / 8\pi G\), where \(\Theta\) is the effective volume, resembles the term \(V \delta P\) in the variation of enthalpy in classical thermodynamics. This suggests that the mass of an AdS black hole should be interpreted as the enthalpy of the spacetime. The paper also discusses the physical interpretation of \(\Theta\) and its role in the first law, and provides a geometric derivation of the Smarr formula and the first law for AdS black holes using the Komar integral relation.