We present a stability result for a wide class of doubly nonlinear equations, featuring general maximal monotone operators and (possibly) nonconvex and nonsmooth energy functionals. The limit analysis consists in the reformulation of the differential evolution as a scalar energy-conservation equation with the aid of the so-called Fitzpatrick theory for the representation of monotone operators. In particular, our result applies to the vanishing viscosity approximation of rate-independent systems.