  The discrepancy between cosmological Li abundance inferred from Population II
dwarf stars and that derived from WMAP/BBNS is still far from being solved.We
investigated, as an alternative route, the use of Li abundances in Population
II lower RGB stars as empirical diagnostic of the cosmological Li. Both theory
and observations suggest that the surface A(Li) in red giants after the
completion of the first dredge-up and before the RGB bump, are significantly
less sensitive to the efficiency of atomic diffusion, compared with dwarf
stars. Standard stellar models computed under different physical assumptions
show that the inclusion of the atomic diffusion has an impact of 0.07dex in the
determination of A(Li)0 (much smaller than the case of MS stars) and it is
basically unaffected by reasonable variations of other parameters
(overshooting, age,initial Y, mixing length). We have determined the surface Li
content of 17 Halo lower RGB stars,in the metallicity range [Fe/H]=-3.4 /-1.4
dex. The initial Li has then been inferred by accounting for the difference
between initial and post-dredge up A(Li) in the appropriate stellar models. It
depends mainly on the used T(eff) scale and is only weakly sensitive to the
efficiency of atomic diffusion,so long as one neglects Li destruction caused by
the process competing with atomic diffusion. Final A(Li)0 span a relatively
narrow range (2.28 /2.46 dex), and is 0.3-0.4 dex lower the WMAP/BBNS
predictions. These values of A(Li)0 are corroborated by the analysis of the GCs
NGC6397, NGC6752 and M4. Our result provides an independent quantitative
estimate of the difference with the Big Bang value and sets a very robust
constraint for the physical processes invoked to resolve this discrepancy.
