  We study the dynamics of gauge-invariant scalar perturbations in cosmological
scenarios with a modified Friedmann equation, such as quantum gravity bouncing
cosmologies. We work within a separate universe approximation which captures
wavelengths larger than the cosmological horizon; this approximation has been
successfully applied to loop quantum cosmology and group field theory. We
consider two variables commonly used to characterise scalar perturbations: the
curvature perturbation on uniform-density hypersurfaces $\zeta$ and the
comoving curvature perturbation $\mathcal{R}$. For standard cosmological models
in general relativity as well as in loop quantum cosmology, these quantities
are conserved and equal on super-horizon scales for adiabatic perturbations.
Here we show that while these statements can be extended to a more general form
of modified Friedmann equations similar to that of loop quantum cosmology, in
other cases, such as the simplest group field theory bounce scenario, $\zeta$
is conserved across the bounce whereas $\mathcal{R}$ is not. We relate our
results to approaches based on a second order equation for a single
perturbation variable, such as the Mukhanov-Sasaki equation.
