  We show that the problem of stabilization of extra dimensions in Kaluza-Klein
type cosmology may be solved in a theory of gravity involving high-order
curvature invariants. The method suggested (employing a slow-change
approximation) can work with rather a general form of the gravitational action.
As examples, we consider pure gravity with Lagrangians quadratic and cubic in
the scalar curvature and some more complex ones in a simple Kaluza-Klein
framework. After a transition to the 4D Einstein conformal frame, this results
in effective scalar field theories with certain effective potentials, which in
many cases possess positive minima providing stable small-size extra
dimensions. Estimates made in the original (Jordan) conformal frame show that
the problem of a small value of the cosmological constant in the present
Universe is softened in this framework but is not solved completely.}
