  Within the Quantum Field Theory context the idea of a "cosmological constant"
(CC) evolving with time looks quite natural as it just reflects the change of
the vacuum energy with the typical energy of the universe. In the particular
frame of Ref.[30], a "running CC" at low energies may arise from generic
quantum effects near the Planck scale, M_P, provided there is a smooth
decoupling of all massive particles below M_P. In this work we further develop
the cosmological consequences of a "running CC" by addressing the accelerated
evolution of the universe within that model. The rate of change of the CC stays
slow, without fine-tuning, and is comparable to H^2 M_P^2. It can be described
by a single parameter, \nu, that can be determined from already planned
experiments using SNe Ia at high z. The range of allowed values for \nu follow
mainly from nucleosynthesis restrictions. Present samples of SNe Ia can not yet
distinguish between a "constant" CC or a "running" one. The numerical
simulations presented in this work show that SNAP can probe the predicted
variation of the CC either ruling out this idea or confirming the evolution
hereafter expected.
