The cohomology of certain families of vector bundles on projective space exhibit a striking stability phenomenon, the proof of which was motivated by similar phenomena for line bundles on flag varieties. In this talk, I'll talk about how this "stable" cohomology is secretly computing derived invariants on the category of polynomial functors, a connection which we can exploit to prove a conjectured representation stability phenomenon for line bundle cohomology. On the other hand, this bridge allows us to apply commutative-algebraic techniques to tackle (and solve) problems in polynomial functor theory. This is based on joint work with Claudiu Raicu.