F-term hybrid inflation models in the context of supergravity generically have corrections to the inflaton mass squared of the order of H2 (the η-problem). In addition they have a problem with large deviations of the spectrum of density perturbations from scale-invariance due to non-renormalizable corrections to the superpotential. Here we show that an increase of the expansion rate at large energy densities relative to that in conventional d = 4 cosmology, as suggested by brane scenarios, can naturally solve the η problem. For the case of a ρ2 squared correction to H2 suppressed by the brane tension this requires that the d = 5 Planck mass satisfies M51016 GeV. In addition, the scale-invariance of the density perturbation spectrum can be generally protected from Planck-suppressed superpotential corrections if M51010{ GeV}. Therefore brane cosmologies which produce an enhanced expansion rate at large energy densities are favoured by SUSY hybrid inflation.