The purpose of this note is to investigate some of the foundational questions concerning convergent cohomology as introduced in [?] and [?], using the language and techniques of Grothendieck topologies. In particular, if X is a scheme of finite type over a perfect field k of characteristic p and with Witt ring W, we define the “convergent topos (X/W)conv,” and we study the cohomology of its structure sheaf OX/W and of KX := Q ⊗ OX/W. Since the topos (X/W)conv is not noetherian, formation of cohomology does not commute with tensor products, and these are potentially quite different.