I might not be sure on this, but the instantaneous partial molar gibbs free energy change at a particular P and T (chemical potential) shows the chemical equivalent of potential energy and thus the graph of it versus the conditions gives us an idea of the "driving force" of the reaction which is maximum if the rate of change of chemical potential is maximum and hence the substance under consideration has the greatest tendency to react or rather to move from one phase to another, i.e equivalent to say that it is less stable.
Partial molar gibbs free energy change is nothing but the change in gibbs free energy per a small change in phase i.e for a small part of reactants to be converted into products in a reaction under consideration. If this is high, slope of the graph of G versus Moles is high the reaction has a greater tendency to move forward
well unfortunately there is no know-it-all thermodynamic function ( atleast that i know of!) because, similar to the variations in stability caused by different pressure and temperature, number of moles and other possible phases in the system also affect the stability. Hence you can only talk about the resistance of of a substance to a particular reaction or a change of phase (in the thermodynamic sense).
For example, methane is quite stable in presence of intense sunlight alone, but if fluorine or chlorine is introduced into the system under consideration, there is explosive conversion of methane into methyl halides and thus the stability is out of question. therefore you would have to compute the potential changes for a mixture with all properties defined before you can comment on its stability.
But as is given in most texts, an almost improper way of understanding stability, is through seeing how negative the heat of formation is but it only gives details as to how stable it is in comparison only to the reactants.