  The fundamental group of the real locus of the Deligne-Mumford
compactification of the moduli space of rational curves with $n$ marked points,
the pure cactus group, resembles the pure braid group in many ways. As it is
the case for several "pure braid like" groups, it is known that its cohomology
ring is generated by its first cohomology. In this note we survey what the
$FI$-module theory developed by Church, Ellenberg and Farb can tell us about
those examples. As a consequence we obtain uniform representation stability for
the sequence of cohomology groups of the pure cactus group.
