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 Friedmann-Robertson-Walker 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 ΛCDM, as the Chevallier-Polarski-Lindler model.