We studied the robustness of a generalized Kitaev's toric code with ℤN degrees of freedom in the presence of local perturbations. For N = 2, this model reduces to the conventional toric code in a uniform magnetic field. A quantitative analysis was performed for the perturbed ℤ3 toric code by applying a combination of high-order series expansions and variational techniques. We found strong evidence for first-and second-order phase transitions between topologically ordered and polarized phases. Most interestingly, our results also indicate the existence of topological multi-critical points in the phase diagram.