Isotropy and homogeneity of space leads to the spacetime metric of the form $$ ds^2=-dt^2+d\sigma_k^2, $$ where $d\sigma_k^2$ is the metric on one of the standard manifolds (the 3-sphere, Euclidean 3-space, and the hyperbolic 3-space) depending on the curvature $k$. Here $k$ may depend on $t$, and there seem to be nothing to prevent $k$ from changing its sign.
However, the Robertson-Walker model assumes $$ ds^2=-dt^2+a(t)^2d\sigma_k^2, $$ where now $k$ is fixed, and the time evolution of the geometry is delegated to the scale factor $a(t)$. This means that the curvature sign-change is forbidden in the Robertson-Walker models.
My question is: Are there cosmological models where the space is allowed to change the sign of its curvature, or is there an argument that it simply cannot happen?