  We study different manifestations of the speed of light in theories of
gravity where metric and connection are regarded as independent fields. We find
that for a generic gravity theory in a frame with locally vanishing affine
connection, the usual degeneracy between different manifestations of the speed
of light is broken. In particular, the space-time causal structure constant
($c_{ST}$) may become variable in that local frame. For theories of the form
$f(R,R^{\mu\nu} R_{\mu\nu})$, this variation in $c_{ST}$ has an impact on the
definition of the luminosity distance (and distance modulus), which can be used
to confront the predictions of particular models against Supernovae type Ia (SN
Ia) data. We carry out this test for a quadratic gravity model without
cosmological constant assuming i) a constant speed of light and ii) a varying
speed of light, and find that the latter scenario is favored by the data.
