The discrepancy between cosmological Li abundance inferred from Population II dwarf stars and that derived from big bang nucleosynthesis calculations is still far from being satisfactorily solved. We investigated, as an alternative route, the use of Li abundances in Population II lower red giant branch stars as empirical diagnostic of the cosmological Li. Both theory and observations suggest that the surface Li abundance in metal-poor red giants after the completion of the first dredge-up and before the red giant branch bump is significantly less sensitive to the efficiency of atomic diffusion, compared with dwarf stars. The surface Li abundances in these objects – after the dilution caused by the first dredge-up – are predicted to be sensitive to the total Li content left in the star, i.e. they are affected only by the total amount of Li eventually burned during the previous main-sequence phase. Standard stellar models computed under different physical assumptions show that the inclusion of the atomic diffusion has an impact of about 0.07 dex in the determination of the primordial Li abundance – much smaller than the case of metal-poor main-sequence turnoff stars – and it is basically unaffected by reasonable variations of other parameters (overshooting, age, initial He abundance and mixing length). We have determined from spectroscopy the surface Li content of 17 halo lower red giant branch stars, in the metallicity range between [Fe/H] ∼− 3.4 and ∼− 1.4 dex, evolving before the extramixing episode that sets in at the red giant branch bump. The initial Li (customarily taken as estimate of the cosmological Li abundance A(Li)0) has then been inferred by accounting for the difference between initial and post-dredge-up Li abundances in the appropriate stellar models. It depends mainly on the Teff scale adopted in the spectroscopic analysis, and is only weakly sensitive to the efficiency of atomic diffusion in the models, so long as one neglects Li destruction caused by the process competing with atomic diffusion. Our final A(Li)0 estimate spans a relatively narrow range, between 2.28 and 2.46 dex, and is ∼0.3–0.4 dex lower than predictions from big bang nucleosynthesis calculations. These values of A(Li)0 are corroborated by the analysis of samples of red giants in the Galactic globular clusters NGC 6397, NGC 6752 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.