  We set up a covariant renormalisation group equation on a foliated spacetime
which preserves background diffeomorphism symmetry. As a first application of
the new formalism, we study the effect of quantum fluctuations in Lorentz
symmetry breaking theories of quantum gravity. It is found that once a small
breaking is introduced e.g. at the Planck scale, quantum fluctuations enhance
this breaking at low energies. A numerical analysis shows that the
magnification is of order unity for trajectories compatible with a small
cosmological constant. The immediate consequence is that the stringent
observational constraints on Lorentz symmetry breaking are essentially
scale-independent and must be met even at the Planck scale.
