We tune the thermodynamics of hydrogen absorption in Mg by means of elastic clamping. The loading isotherms measured by hydrogenography show that Mg films covered with Mg-alloy-forming elements, such as Pd and Ni, have hydrogen plateau pressures more than 2 orders of magnitude higher than bulk Mg at the same temperature. An elastic model allows us to interpret the Mg thickness dependence of the hydrogen plateau pressure. Our results suggest an alternative route for the development of new hydrogen storage materials with optimized thermodynamic properties.