  We have studied the lattice dynamics, electron-phonon coupling, and
superconducting properties of $\alpha$-MoB$_2$, as a function of applied
pressure, within the framework of density functional perturbation theory using
a mixed-basis pseudopotential method. We found that phonon modes located along
the A$-$H, H$-$L, and L$-$A high-symmetry paths exhibit large phonon linewidths
and contribute significantly to the electron-phonon coupling constant. Although
linewidths are particularly large for the highest-frequency optical phonon
modes (dominated by B vibrations), their contribution to the electron-phonon
coupling constant is marginal. The latter is largely controlled by the acoustic
low-frequency modes of predominantly Mo character. It was observed that at a
pressure of $90$~GPa, where $\alpha$-MoB$_2$ forms, the phonon-mediated pairing
falls into the strong-coupling regime, and the estimate for the superconducting
critical temperature $T_c$ agrees well with experimental observations. When
further increasing the applied pressure, a reduction of $T_c$ is predicted,
which correlates with a hardening of the acoustic low-frequency phonon modes
and a decrease of the electron-phonon coupling parameter.
