This work satisfies cohomology theory within the framework of fuzzy bornological groups. The primary objective is to establish a robust formalization of cohomology for these fuzzy groups and to extend the fundamental properties of classical cohomology to the fuzzy group setting. To achieve this, we construct a cohomology theory for a fuzzy bornological group (G ̃,δ ̃) and a fuzzy bornological G ̃-modules (M ̃  ,δ ̃), by utilizing the concept of fuzzy bounded cochains.