Riemann tensor and Gauss–Bonnet density in metric-affine cosmology

We analytically derive the covariant form of Riemann (curvature) tensor for homogeneous metric-affine cosmologies. That is, we present, in a cosmological setting, most general full including also its non-Riemannian pieces which are associated to spacetime torsion and non-metricity. Having done so compute list curvature by-products such as Ricci tensor, homothetic curvature, scalar, Einstein etc. Finally generalized version usual Gauss–Bonnet density this background demonstrate how under certain circumstances latter represents total derivative term.