The cosmological lithium problem, varying constants and the $H_0$ tension

In this work we show that the cosmological lithium problem and the $H_0$ tension could be eased at the same time by allowing variations in the fundamental constants. We compute the primordial abundances of light elements resulting from Big Bang Nucleosynthesis considering the fine structure constant, the Higgs' vacuum expectation value and Newton's constant as free parameters. Using the observational data for abundances, we set constraints on the variations of the fundamental constants. An interpretation of the results in terms of the number of effective relativistic species gives $N_{\rm eff}=4.04\pm0.12$. If one extrapolates the fit of {\sffamily Planck} considering this $N_{\rm eff}$, the value of the inferred Hubble's constant shifts to $H_0=(71.85\pm0.77)\,\text{km}\,\text{s}^{-1} \text{Mpc}^{-1}$, compatible with current direct determinations.