The local Pair (LP) superconductor is investigated by means of a Monte Carlo (MC) method. The ground state phase-diagram in the density versus repulsion/transfer integral plane is calculated, and compared to analytical (RPA) results. The phase diagram features the superconducting phase, a charge-ordered phase and a region where superconductivity and charge ordering coexist. Also, the finite temperature phase diagram and the temperature dependences of the specific heat and the superconducting order parameter are presented.