We show that the orbital angular momentum (OAM) of a light field can be thermalized in a nonlinear cylindrical multimode optical waveguide. We find that upon thermal equilibrium, the maximization of the optical entropy leads to a generalized Rayleigh-Jeans distribution that governs the power modal occupancies with respect to the discrete OAM charge numbers. This distribution is characterized by a temperature that is by nature different from that associated with the longitudinal electromagnetic momentum flow of the optical field. Counterintuitively and in contrast to previous results, we demonstrate that even under positive temperatures, the ground state of the fiber is not always the most populated in terms of power. Instead, because of OAM, the thermalization processes may favor higher-order modes. A new equation of state is derived along with an extended Euler equation resulting from the extensivity of the entropy itself. By monitoring the nonlinear interaction between two multimode optical wave fronts with opposite spins, we show that the exchange of angular momentum is dictated by the difference in OAM temperatures, in full accord with the second law of thermodynamics. The theoretical analysis presented here is corroborated by numerical simulations that take into account the complex nonlinear dynamics of hundreds of modes. Our results may pave the way toward high-power optical sources with controllable orbital angular momenta, and at a more fundamental level, they may open up opportunities in drawing parallels with other complex multimode nonlinear systems like rotating atomic clouds.