  We examine the general issue of whether a scale dependent cosmological
constant can be consistent with general covariance, a problem that arises
naturally in the treatment of quantum gravitation where coupling constants
generally run as a consequence of renormalization group effects. The issue is
approached from several points of view, which include the manifestly covariant
functional integral formulation, covariant continuum perturbation theory about
two dimensions, the lattice formulation of gravity, and the non-local effective
action and effective field equation methods. In all cases we find that the
cosmological constant cannot run with scale, unless general covariance is
explicitly broken by the regularization procedure. Our results are expected to
have some bearing on current quantum gravity calculations, but more generally
should apply to phenomenological approaches to the cosmological vacuum energy
problem.
