The production of $\phi$ 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 $\mathrm{d}^2\sigma/$\mathrm{d}y $\mathrm{d} p_\mathrm{T}$ are presented as a function of the transverse momentum ($p_\mathrm{T}$) at the center-of-mass energies $\sqrt{s}=5.02$, 8 and 13 TeV and compared with the ALICE results at midrapidity. The differential cross sections at $\sqrt{s}=5.02$ and 13 TeV are also studied in several rapidity intervals as a function of $p_\mathrm{T}$, and as a function of rapidity in three $p_\mathrm{T}$ intervals. A hardening of the $p_\mathrm{T}$-differential cross section with the collision energy is observed, while, for a given energy, $p_\mathrm{T}$ spectra soften with increasing rapidity and, conversely, rapidity distributions get slightly narrower at increasing $p_\mathrm{T}$. The new results, complementing the published measurements at $\sqrt{s}=2.76$ and 7 TeV, allow one to establish the energy dependence of $\phi$ meson production and to compare the measured cross sections with phenomenological models.

$\phi$ meson production cross section $\mathrm{d}^2\sigma/(\mathrm{d}y\mathrm{d}p_\mathrm{T})$ as a function of $p_\mathrm{T}$ at $\sqrt{s}=5.02$ TeV in pp collisions, in the $2.5