  In this paper, we investigate the coexistence curves and molecule number
densities of $f(R)$ AdS black holes and Gauss-Bonnet AdS black holes.
Specifically, we work with the reduced parameter space and derive the analytic
expressions of the universal coexistence curves that are independent of theory
parameters. Moreover, we obtain the explicit expressions of the physical
quantity describing the difference of the number densities of black hole
molecules between the small and large black hole. It is found that both the
coexistence 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 attributed
to the same equation of state they share in the reduced parameter space. The
difference of the molecule number densities between the small and large
Gauss-Bonnet AdS black hole exhibits different behavior. This may be attributed
to the fact that the charge of RN-AdS black hole is non-trivial. Our research
will not only deepen the understanding of both the physics along the
coexistence curve and the underlying microscopic freedom of AdS black holes,
but also highlight the importance of the law of corresponding states.
