Subject of the talk is a multidimensional gravitational model with scalar curvature nonlinearity R^4. It is assumed that the higher dimensional spacetime manifold of this model undergoes a spontaneous compactification to a manifold with warped product structure. The main attention is paid to the stability of the extra-dimensional factor spaces and it is shown that for certain parameter regions the system allows for a freezing stabilization of these spaces. The most interesting fact which is demonstrated is a dependence of the stability region (in parameter space) on the total dimension D=dim(M) of the higher dimensional spacetime M. For D>8 the stability region consists of a single (absolutely stable) sector which is shielded from a conformal singularity (and an antigravity sector beyond it) by a potential barrier of infinite height and width. This sector is smoothly connected with the stability region of a curvature-linear model. For D<8 an additional (metastable) sector exists which is separated from the conformal singularity by a potential barrier of finite height and width so that systems in this sector are prone to collapse into the conformal singularity. This second sector is not smoothly connected with the first (absolutely stable) one. Several limiting cases and the possibility for inflation are discussed. (hep-th/0409112)