In latitude-dependent energy balance models, ice-free and ice-covered conditions form physical boundaries of the system. With carbon dioxide treated as a bifurcation parameter, the resulting bifurcation diagram is nonsmooth with curves of equilibria and boundaries forming corners at points of intersection. Over long time scales, atmospheric carbon dioxide varies dynamically and the nonsmooth diagram becomes a set of quasi-equilibria. In this article, we extend an energy balance model to include slowly varying carbon dynamics in a way that allows for a nonsmooth dynamical systems treatment. In our analysis within this framework, we prove existence and forward uniqueness of solutions and show that the physical region is forward invariant with sliding motion on the physical boundaries.