  We give a characterization of those Alexandrov spaces admitting a
cohomogeneity one action of a compact connected Lie group $G$ for which the
action is Cohen--Macaulay. This generalizes a similar result for manifolds to
the singular setting of Alexandrov spaces where, in contrast to the manifold
case, we find several actions which are not Cohen--Macaulay. In fact, we
present results in a slightly more general context. We extend the methods in
this field by a conceptual approach on equivariant cohomology via rational
homotopy theory using an explicit rational model for a double mapping cylinder.
