The absolute HOMO and LUMO energies will certainly be related to susceptibility of a substance to oxidation and reduction processes because HOMOs and LUMOs tend to be correlated to redox potentials. E.g. a higher HOMO energy means that a substance is easier to oxidize. If the HOMO energy is high enough, a substance may not be air stable. The HOMO-LUMO energy gap is also related to certain photophysical or even thermal processes that could impact stability because photoexcited states tend to be more reactive than ground state analogs.
That said, these relationships would be complex and may not lend themselves well to a simplistic analysis of "HOMO-LUMO gap versus stability". You'd have to have a metric for stability for one thing. For another, stability will depend on a lot things such as the environment composition, temperature, and so forth. In the case of photo-stability, you also need to have the appropriate light wavelengths to bring about deleterious photochemistry.