  We study the magnetoelectric and electrocaloric response of
strain-engineered, multiferroic SrMnO$_3$, using a phenomenological Landau
theory with all parameters obtained from \emph{first-principles}-based
calculations. This allows to make realistic and materials-specific predictions
about the magnitude of the corresponding effects. We find that in the vicinity
of a tetracritical point, where magnetic and ferroelectric phase boundaries
intersect, an electric field has a huge effect on the antiferromagnetic order,
corresponding to a magnetoelectric response several orders of magnitude larger
than in conventional linear magnetoelectrics. Furthermore, the strong
magnetoelectric coupling leads to a magnetic, cross-caloric contribution to the
electrocaloric effect, which increases the overall caloric response by about
60\%. This opens up new potential applications of antiferromagnetic
multiferroics in the context of environmentally friendly solid state cooling
technologies.
