  We consider spatially homogeneous (but generally non-isotropic) cosmologies
in the recently proposed Horava-Lifshitz gravity and compare them to those of
general relativity using Hamiltonian methods. In all cases, the problem is
described by an effective point particle moving in a potential well with
exponentially steep walls. Focusing on the closed-space cosmological model
(Bianchi type IX), the mixmaster dynamics is now completely dominated by the
quadratic Cotton tensor potential term for very small volume of the universe.
Unlike general relativity, where the evolution towards the initial singularity
always exhibits chaotic behavior with alternating Kasner epochs, the
anisotropic universe in Horava-Lifshitz gravity (with parameter lambda > 1/3)
is described by a particle moving in a frozen potential well with fixed (but
arbitrary) energy E. Alternating Kasner epochs still provide a good description
of the early universe for very large E, but the evolution appears to be
non-ergodic. For very small E there are harmonic oscillations around the fully
isotropic model. The question of chaos remains open for intermediate energy
levels.
