  We show that a number of problems of modern cosmology may be solved in the
framework of multidimensional gravity with high-order curvature invariants,
without invoking other fields. We use a method employing a slow-change
approximation, able to work with rather a general form of the gravitational
action, and consider Kaluza-Klein type space-times with one or several extra
factor spaces. A vast choice of effective theories suggested by the present
framework may be stressed: even if the initial Lagrangian is entirely fixed,
one obtains quite different models for different numbers, dimensions and
topologies of the extra factor spaces. As examples of problems addressed we
consider (i) explanation of the present accelerated expansion of the Universe,
with a reasonably small cosmological constant, and the problem of its fine
tuning is considered from a new point of view; (ii) the mechanism of closed
wall production in the early Universe; such walls are necessary for massive
primordial black hole formation which is an important stage in some scenarios
of cosmic structure formation; (iii) sufficient particle production rate at the
end of inflation; (iv) it is shown that our Universe may contain spatial
domains with a macroscopic size of extra dimensions. We also discuss chaotic
attractors appearing at possible nodes of the kinetic term of the effective
scalar field Lagrangian.
