Stability analysis of black holes via a catastrophe theory and black hole thermodynamics in generalized theories of gravity

We perform a linear perturbation analysis for black hole solutions with a "massive" Yang-Mills field (the Proca field) in Brans-Dicke theory and find that the results are quite consistent with those via catastrophe theory where thermodynamic variables play an intrinsic role. Based on this observation, we show the general relation between these two methods in generalized theories of gravity which are conformally related to the Einstein-Hilbert action.