Abstract Understanding when global glaciations occur on Earth-like planets is a major challenge in climate evolution research. Most models of how greenhouse gases like CO 2 evolve with time on terrestrial planets are deterministic, but the complex, nonlinear nature of Earth’s climate history motivates study of nondeterministic climate models. Here a maximally simple stochastic model of CO 2 evolution and climate on an Earth-like planet with an imperfect CO 2 thermostat is investigated. It is shown that as stellar luminosity is increased in this model, the decrease in the average atmospheric CO 2 concentration renders the climate increasingly unstable, with excursions to a low-temperature state common once the received stellar flux approaches that of present-day Earth. Unless climate feedbacks always force the variance in CO 2 concentration to decline rapidly with received stellar flux, this means that terrestrial planets near the inner edge of the habitable zone may enter Snowball states quite frequently. Observations of the albedos and color variation of terrestrial-type exoplanets should allow this prediction to be tested directly in the future.