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In this work it is introduced an extension of the study of the first law ofthermodynamics of black holes based on the geometry of the extended phase spacefor AdS Lovelock gravities which includes changes of scales. As expected, theresult obtained coincides with the previously known four dimensional case.  For higher dimensions, the result is the arise of two new contributions tothe first law of thermodynamics. The first term corresponds to corrections ofthe usual definition of thermodynamics volume at the horizon, due to thepresence of the higher curvature terms. The second term arises in odddimensions, comes from the asymptotic region, and corresponds to a scaletransformation of the form $\propto \hat{\delta} \ln (l/\ell)$, with $l$ theAdS radius and $\ell$ a parameter. A particularly interesting case correspondsto the Chern Simons gravity where the change scale does not generate acontribution at the asymptotic region, likely due to the Chern Simons AdS localsymmetry.

An alternative study about the geometry and the first law of thermodynamics for AdS Lovelock gravity, using the definition of conserved charges