  In this paper we propose a scheme which allows one to find all possible
exponential solutions of special class -- non-constant volume solutions -- in
Lovelock gravity in arbitrary number of dimensions and with arbitrate
combinations of Lovelock terms. We apply this scheme to (6+1)- and
(7+1)-dimensional flat anisotropic cosmologies in Einstein-Gauss-Bonnet and
third-order Lovelock gravity to demonstrate how our scheme does work. In course
of this demonstration we derive all possible solutions in (6+1) and (7+1)
dimensions and compare solutions and their abundance between cases with
different Lovelock terms present. As a special but more "physical" case we
consider spaces which allow three-dimensional isotropic subspace for they could
be viewed as examples of compactification schemes. Our results suggest that the
same solution with three-dimensional isotropic subspace is more "probable" to
occur in the model with most possible Lovelock terms taken into account, which
could be used as kind of anthropic argument for consideration of Lovelock and
other higher-order gravity models in multidimensional cosmologies.
