The nonequilibrium dynamics of light in a coherently driven nonlinear cavity resembles the equilibrium dynamics of a Brownian particle in a scalar potential. This resemblance has been known for decades, but the correspondence between the two systems has never been properly assessed. Here we demonstrate that this correspondence can be exact, be approximate, or break down, depending on the driving conditions. For vanishing nonlinearity and on-resonance driving, the correspondence is exact: The cavity dissipation and driving amplitude define a scalar potential, and light follows the equilibrium Boltzmann distribution with an effective temperature defined by the noise variance and cavity dissipation. The scalar potential pertaining to linear on-resonance dynamics fails dramatically in nonlinear and/or off-resonance regimes. However, we introduce a distinct scalar potential enabling an effective equilibrium description of light. Our potential gives a reasonably accurate description in limited nonlinear regimes of bistability, but fails deep in the bistability where nonconservative forces dominate the dynamics. Consequently, the correspondence to Brownian motion in a scalar potential breaks down. This breakdown is accompanied by a qualitative change in the spectrum of small intracavity field fluctuations, reminiscent of an exceptional point of a non-Hermitian Hamiltonian. Our results lay the foundations for an effective thermodynamic description of coherently driven cavities, and suggest that fundamental results for overdamped Langevin dynamics can help to assess the energetics and information processing of resonant optical technologies.