In most of the investigations on metastable systems, the stochastic forcing is modulated by Gaussian noise. Lévy noise laws, which describe jump processes, have recently received a lot of attention, but much less is known. We study stochastic versions of the Ghil–Sellers energy balance model, and we highlight the fundamental difference between how transitions are performed between the competing warm and snowball states, depending on whether Gaussian or Lévy noise acts as forcing.