  The thermal freeze-out mechanism for relic dark matter heavier than $O(10-100
$ TeV$)$ requires cross-sections that violate perturbative unitarity. Yet the
existence of dark matter heavier than these scales is certainly plausible from
a particle physics perspective, pointing to the need for a non-thermal
cosmological history for such theories. Topological dark matter is a
well-motivated scenario of this kind. Here the hidden-sector dark matter can be
produced in abundance through the Kibble-Zurek mechanism describing the
non-equilibrium dynamics of defects produced in a second order phase
transition. We revisit the original topological dark matter scenario, focusing
on hidden-sector magnetic monopoles, and consider more general cosmological
histories. We find that a monopole mass of order ($1-10^5$) PeV is generic for
the thermal histories considered here, if monopoles are to entirely reproduce
the current abundance of dark matter. In particular, in a scenario involving an
early era of matter domination, the monopole number density is always less than
or equal to that in a pure radiation dominated equivalent provided a certain
condition on critical exponents is satisfied. This results in a larger monopole
mass needed to account for a fixed relic abundance in such cosmologies.
