The magnitude of the double longitudinal spin asymmetry of π-, K-, and η-mesons generated in collisions of longitudinally polarized protons at energies of 510, 200, and 27 GeV has been analyzed (using Monte Carlo simulation). The simulation was carried out using the Pythia8 software package, which had been previously modified for the task at hand: accounting for spiral parton distribution functions and interaction cross-sections of polarized partons with certain spiralities. A comparison of the simulation results with experimental data for energy values of 510 and 200 GeV from literature sources showed a good agreement. This served as the basis for calculating the results of the planned NICA experiment for the considered mesons at an energy of 27 GeV.