  The highest superconducting temperature T$_c$ observed in any elemental metal
(Li with T$_c$ ~ 20 K at pressure P ~ 40 GPa) is shown to arise from critical
(formally divergent) electron-phonon coupling to the transverse T$_1$ phonon
branch along intersections of Kohn anomaly surfaces with the Fermi surface.
First principles linear response calculations of the phonon spectrum and
spectral function $\alpha^2 F(\omega)$ reveal (harmonic) instability already at
25 GPa. Our results imply that the fcc phase is anharmonically stabilized in
the 25-38 GPa range.
