In this work we perform a computational study comparing the influence of monovalent cation substitution by methylammonium (MA(+)), cesium (Cs(+)), and rubidium (Rb(+)) on the properties of formamidinium lead triiodide (FAPbI3)-based perovskites. The relative stability of the desired, photoactive perovskite α phase ("black phase") and the nonphotoactive, nonperovskite δ phase ("yellow phase") is studied as a function of dopant nature, concentration and temperature. Cs(+) and Rb(+) are shown to be more efficient in the stabilization of the perovskite α phase than MA(+). Furthermore, varying the dopant concentration allows changing the relative stability at different temperatures, in particular stabilizing the α phase already at 200 K. Upon Cs(+) or Rb(+) doping, the corresponding onset of the optical spectrum is blue-shifted by 0.1-0.2 eV with respect to pure FAPbI3.