Being not at all a specialist of the subject my question is maybe naive, I apologize if it's the case. The question is the following: There are many different version of entropy functionals around (Boltzmann, Tsallis and many others...). Usually they satisfy the first 3 SHannon-Khinchin axioms but the 4th, due to the uniquess theorem of Shannon-Khinchin, is some sort of twisted version of composability like $$S(AB) = S(A) + S(B) + (some other term) .$$ (In case of Shannon entropy we know that $$S(AB) = S(A) + S(B)).$$ There are also various notions of composability around (see e.g. G. Tempesta). MY question is the following:
IS it physically interesting or even reasonable (do we have examples of physical situations?) to have a "sub-composable" entropy S in the sens that $$S(AB) \leq S(A) + S(B)~?$$