  Horava-Lifshitz theory of gravity with detailed balance is plagued by the
presence of a negative bare (or geometrical) cosmological constant which makes
its cosmology clash with observations. We argue that adding the effects of the
large vacuum energy of quantum matter fields, this bare cosmological constant
can be approximately compensated to account for the small observed (total)
cosmological constant. Even though we cannot address the fine-tuning problem in
this way, we are able to establish a relation between the smallness of observed
cosmological constant and the length scale at which dimension 4 corrections to
the Einstein gravity become significant for cosmology. This scale turns out to
be approximately 5 times the Planck length for an (almost) vanishing observed
cosmological constant and we therefore argue that its smallness guarantees that
Lorentz invariance is broken only at very small scales. We are also able to
provide a first rough estimation for the infrared values of the parameters of
the theory $\mu$ and $Lambda_w$.
