The long-standing prediction that hydrogen can assume a metallic state under high pressure, combined with arguments put forward more recently that this state might even be superconducting up to high temperatures, continues to spur tremendous research activities toward the experimental realization of metallic hydrogen. These efforts have however so far been impeded by the enormous challenges associated with the exceedingly large required pressure. Hydrogen-dense materials, of the MH₄ form (where M can be, e.g., Si, Ge, or Sn) or of the MH₃ form (with M being, e.g., Al, Sc, Y, or La), allow for the rather exciting opportunity to carry out a proxy study of metallic hydrogen and associated high-temperature superconductivity at pressures within the reach of current techniques. At least one experimental report indicates that a superconducting state might have been observed already in SiH₄, and several theoretical studies have predicted superconductivity in pressurized hydrogen-rich materials; however, no systematic dependence on the applied pressure has yet been identified so far. In the present work, we have used first-principles methods in an attempt to predict the superconducting critical temperature (T_c) as a function of pressure (P) for three metal-hydride systems of the MH₃ form, namely ScH₃, YH₃, and LaH₃. By comparing the obtained results, we are able to point out a general trend in the T_c-dependence on P. These gained insights presented here are likely to stimulate further theoretical studies of metallic phases of hydrogen-dense materials and should lead to new experimental investigations of their superconducting properties.