We study the magnetoelectric and electrocaloric response of strain-engineered, multiferroic SrMnO₃, using a phenomenological Landau theory with all parameters obtained from first-principles-based calculations. This allows to make realistic semi-quantitative 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.