Friedmann equations in the Codazzi parametrization of Cotton and extended theories of gravity and the Dark Sector

The Friedmann equations of Cotton gravity provide a simple parametrization to reproduce, by tuning a single function, the Friedmann equations of several extensions of gravity, such as f(R), modified Gauss-Bonnet f(G), teleparallel f(T), and more. It also includes the recently proposed Conformal Killing gravity and Mimetic gravity in FRW space-times. The extensions generally have the form of a Codazzi tensor that may be associated to the dark sector. Fixing it by a suitable equation of state accomodates most of the postulated models that extend $\Lambda$CDM, as the Chevallier-Polarski-Lindler model.
Comment: Accepted in PRD. Generic Gravity in section 3 and new references