  We present a first-principles study of the electronic structure and
superconducting parameters of the compounds $A$Os$_2$O$_6$ ($A$ = Na, K, Rb,
and Cs) and at ambient and applied hydrostatic pressure. We find that the
sensitivity of the density of states at the Fermi energy, $E_{\rm F}$, and
related electronic properties to the size of the alkali metal atom as well as
to applied pressure is driven by a van Hove singularity with energy very close
to $E_{\rm F}$. Further, a computation of the superconducting parameters of
these materials allows us to show that the observed change of $T_c$, both upon
substitution of the alkali metal and under applied hydrostatic pressure, can be
well understood within a phonon-mediated pairing scenario. In this regard, we
find that the correction to the effective electron mass due to spin
fluctuations plays a significant role.
