  An interesting phenomenological consequence of Lambda varying gravity
theories inspired by quantum gravity models is reported. The treatment in the
present work is quite general and applicable to several different actions with
Lambda varying, especially those used in RG approaches to quantum gravity. An
effective gravitational action with a scale varying cosmological constant,
Lambda, which depends on the system's characteristics, like the length and the
energy density, is the key feature. If the system is an astrophysical object,
like a cluster of galaxies, a black hole, etc, non-negligible corrections arise
to several observable quantities. Distinctive footprints could refer to
luminosity distance and strong/weak lensing measurements, among others. The
present study focuses on the SNIa luminosity distance observable.
