Summary: The purpose of this note is twofold. Firstly, the authors consider generalizations of Shannon’s entropy and its applications to thermodynamics based on extensivity considerations. Secondly, they apply the generalized entropy formalism to deriving various generalised channel capacities. They arrive at some surprising conclusions of systems achieving “super-capacitance” or “sub-capacitance” depending on the circumstances. These results suggest the possibility of improving the conventional Shannon capacity by using physical systems obeying more generalized statistics, but also predict behaviour which is hard to reconcile with experience.