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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 (FAPbI(3))based perovskites. The relative stability of the desired, photoactive perovskite alpha phase ("black phase") and the nonphotoactive, nonperovskite delta 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 alpha phase than MA(+). Furthermore, varying the dopant concentration allows changing the relative stability at different temperatures, in particular stabilizing the alpha 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 FAPbI(3)