  Motivated by the absence of experimental superconductivity in the metallic
Pm-3n phase of AlH3 despite the predictions, we reanalyze its vibrational and
superconducting properties at pressures above 99 GPa making use of
first-principles techniques. In our calculations based on the self-consistent
harmonic approximation method that treats anharmonicity beyond perturbation
theory, we predict a strong anharmonic correction to the phonon spectra and
demonstrate that the superconducting critical temperatures predicted in
previous calculations based on the harmonic approximation are strongly
suppressed by anharmonicity. The electron-phonon coupling concentrates on the
lowest-energy hydrogen-character optical modes at the X point of the Brillouin
zone. As a consequence of the strong anharmonic enhancement of their frequency,
the electron-phonon coupling is suppressed by at least a 30%. The suppression
in {\lambda} makes Tc smaller than 4.2 K above 120 GPa, which is well
consistent with the experimental evidence. Our results underline that metal
hydrides with hydrogen atoms in interstitial sites are subject to huge
anharmonic effects.
