Let n be a positive integer and X a scheme. Then for all the Grothendieck topologies I know (Zariski, etale, fppf) the set $H^1(X,GL_n)$ is the set of (isomorphism classes of) rank $n$ vector bundles.
Is there a Grothendieck topology (something not too trivial...) so that the corresponding H^1 is different? (Of course, you will need GL_n to still be a sheaf.)
I'm thinking about Nisnevich/Voevodksy type topologies for example.