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In this paper, we investigate the coexistence curves and molecule numberdensities of $f(R)$ AdS black holes and Gauss-Bonnet AdS black holes.Specifically, we work with the reduced parameter space and derive the analyticexpressions of the universal coexistence curves that are independent of theoryparameters. Moreover, we obtain the explicit expressions of the physicalquantity describing the difference of the number densities of black holemolecules between the small and large black hole. It is found that both thecoexistence curve and the difference of the molecule number densities of $f(R)$AdS black holes coincide with those of RN-AdS black holes. It may be attributedto the same equation of state they share in the reduced parameter space. Thedifference of the molecule number densities between the small and largeGauss-Bonnet AdS black hole exhibits different behavior. This may be attributedto the fact that the charge of RN-AdS black hole is non-trivial. Our researchwill not only deepen the understanding of both the physics along thecoexistence curve and the underlying microscopic freedom of AdS black holes,but also highlight the importance of the law of corresponding states.

Coexistence curves and molecule number densities of AdS black holes in the reduced parameter space