Context. Some models of the formation of first galaxies predict low masses and faint objects at extremely high redshifts, z ≃ 9 − 15. However, the first observations of this epoch indicate a higher-than-expected number of bright (sometimes massive) galaxies. Aims. Numerical simulations can help to elucidate the mild evolution of the bright end of the UV luminosity function and they can provide the link between the evolution of bright galaxies and variations of the galaxy formation efficiency across different redshifts. Methods. We use the FirstLight database of 377 zoom-in cosmological simulations of a volume- and mass-complete sample of galaxies. Mock luminosities are estimated by a dust model constrained by observations of the β–MUV relation at z = 6 − 9. Results. FirstLight contains a high number of bright galaxies, MUV ≤ −20, consistent with current data at z = 6 − 13. The evolution of the UV cosmic density is driven by the evolution of the galaxy efficiency and the relation between MUV and halo mass. The efficiency of galaxy formation increases significantly with mass and redshift. At a fixed mass, galactic halos at extremely high redshifts convert gas into stars at a higher rate than at lower redshifts. The high gas densities in these galaxies enable high efficiencies. Our simulations predict higher number densities of massive galaxies, M* ≃ 109 M⊙, than other models with constant efficiency. Conclusions. Cosmological simulations of galaxy formation with detailed models of star formation and feedback can reproduce the different regimes of galaxy formation across cosmic history.