The production of ϕ mesons has been studied in pp collisions at LHC energies with the ALICE detector via the dimuon decay channel in the rapidity region 2.5 < y< 4. Measurements of the differential cross section d ²σ/ d yd p_T are presented as a function of the transverse momentum (p_T) at the center-of-mass energies s=5.02, 8 and 13 TeV and compared with the ALICE results at midrapidity. The differential cross sections at s=5.02 and 13 TeV are also studied in several rapidity intervals as a function of p_T, and as a function of rapidity in three p_T intervals. A hardening of the p_T-differential cross section with the collision energy is observed, while, for a given energy, p_T spectra soften with increasing rapidity and, conversely, rapidity distributions get slightly narrower at increasing p_T. The new results, complementing the published measurements at s=2.76 and 7 TeV, allow one to establish the energy dependence of ϕ meson production and to compare the measured cross sections with phenomenological models. None of the considered models manages to describe the evolution of the cross section with p_T and rapidity at all the energies.