  The minimal requirement for cosmography - a nondynamical description of the
universe - is a prescription for calculating null geodesics, and timelike
geodesics as a function of their proper time. In this paper, we consider the
most general linear connection compatible with homogeneity and isotropy, but
not necessarily with a metric. A light-cone structure is assigned by choosing a
set of geodesics representing light rays. This defines a "scale factor" and a
local notion of distance, as that travelled by light in a given proper time
interval. We find that the velocities and relativistic energies of free-falling
bodies decrease in time as a consequence of cosmic expansion, but at a rate
that can be different than that dictated by the usual metric framework. By
extrapolating this behavior to photons redshift, we find that the latter is in
principle independent of the "scale factor". Interestingly, redshift-distance
relations and other standard geometric observables are modified in this
extended framework, in a way that could be experimentally tested. An extremely
tight constraint on the model, however, is represented by the blackbody-ness of
the Cosmic Microwave Background. Finally, as a check, we also consider the
effects of a non-metric connection in a different set-up, namely, that of a
static, spherically symmetric spacetime.
