  Within the theoretical framework of some modern unification theories the
constants of nature are functions of cosmological time. White dwarfs offer the
possibility of testing a possible variation of G and, thus, to place
constraints to these theories. We present full white dwarf evolutionary
calculations in the case that G decreases with time. White dwarf evolution is
computed in a self-consistent way, including the most up-to-date physical
inputs, non-gray model atmospheres and a detailed core chemical composition
that results from the calculation of the full evolution of progenitor stars. We
find that the mechanical structure and the energy balance of white dwarfs are
strongly modified by the presence of a varying G. In particular, for certain
values of the rate of change of G, the evolution of cool white dwarfs is
markedly affected. The impact of a varying G is more notorious in the case of
more massive white dwarfs. In view of the recent results reporting that a very
accurate white dwarf cooling age can be derived for the old and metal-rich open
cluster NGC 6791, our study suggests that this cluster could be a potential
target to constrain or detect a ypothetical secular variation of G.
