Climate variability has significant human impact but is difficult to model and predict. Recent theoretical developments in nonequilibrium statistical mechanics cover a class of stochastic models often used for climate phenomena. The theory for a coarse-grained entropy production is developed for simple stochastic climate models and applied to observed tropical sea surface temperatures (SST), demonstrating that nonequilibrium properties can be quantified with climate datasets, and that tropical SST variability is approximately consistent with fluctuations about a nonequilibrium steady-state with relatively few degrees-of-freedom. Fluctuations with negative entropy production demonstrate that seasonal tropical SST variability is small and fast in a thermodynamic sense, indicating that nonequilibrium fluctuation theory is applicable. This work demonstrates that nonequilibrium theory can address climate-scale phenomena, suggests that it could provide insight into how climate change will affect climate variability, and perhaps provide a fundamental theory for variability of the climate system.