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In this paper we address two important issues which could affect reaching theexponential and Kasner asymptotes in Einstein-Gauss-Bonnet cosmologies --spatial curvature and anisotropy in both three- and extra-dimensionalsubspaces. In the first part of the paper we consider cosmological evolution ofspaces being the product of two isotropic and spatially curved subspaces. It isdemonstrated that the dynamics in $D=2$ (the number of extra dimensions) and $D\geqslant 3$ is different. It was already known that for the $\Lambda$-termcase there is a regime with "stabilization" of extra dimensions, where theexpansion rate of the three-dimensional subspace as well as the scale factor(the "size") associated with extra dimensions reach constant value. This regimeis achieved if the curvature of the extra dimensions is negative. Wedemonstrate that it take place only if the number of extra dimensions is $D\geqslant 3$. In the second part of the paper we study the influence of initialanisotropy. Our study reveals that the transition from Gauss-Bonnet Kasnerregime to anisotropic exponential expansion (with expanding three andcontracting extra dimensions) is stable with respect to breaking the symmetrywithin both three- and extra-dimensional subspaces. However, the details of thedynamics in $D=2$ and $D \geqslant 3$ are different. Combining the twodescribed affects allows us to construct a scenario in $D \geqslant 3$, whereisotropisation of outer and inner subspaces is reached dynamically from rathergeneral anisotropic initial conditions.

Effects of spatial curvature and anisotropy on the asymptotic regimes in Einstein-Gauss-Bonnet gravity