  We use a combination of density functional theory and Monte Carlo
calculations to calculate the surface magnetization in magnetoelectric
$\mathrm{Cr_2O_3}$ at finite temperatures. Such antiferromagnets, lacking both
inversion and time-reversal symmetries, are required by symmetry to posses an
uncompensated magnetization density on particular surface terminations. Here,
we first show that the uppermost layer of magnetic moments on the $(001)$
surface remain paramagnetic at the bulk N\'{e}el temperature, bringing the
theoretical estimate of surface magnetization density in line with experiment.
We demonstrate that the lower surface ordering temperature compared to bulk is
a generic feature of surface magnetization when the termination reduces the
effective Heisenberg coupling. We then propose two methods by which the surface
magnetization in $\mathrm{Cr_2O_3}$ could be stabilised at higher temperatures.
Specifically, we show that the effective coupling of surface magnetic ions can
be drastically increased either by a different choice of surface Miller plane,
or by $\mathrm{Fe}$ doping. Our findings provide an improved understanding of
surface magnetization properties in AFMs.
