The majority of results in the literature on general equilibrium are not for an economy (i.e. given an endowment and preferences), but rather, for a set of economies (i.e. a set of endowments given preferences). Therefore, we argue that the most appropriate robustness result requires perturbing economies uniformly over the space of endowments for which the result is obtained. In this paper, we examine the robustness of the uniqueness of Walrasian endowment economies with Cobb-Douglas utility functions under this interpretation of robustness. Namely, we prove that for economies described by Cobb-Douglas utilities and all endowments in a fixed set, uniqueness of equilibrium is robust to perturbations of the utility functions.