  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 ${\rm
d}^2\sigma/{\rm d}y {\rm d}p_{\rm T}$ are presented as a function of the
transverse momentum ($p_{\rm T}$) 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 $p_{\rm T}$, and as
a function of rapidity in three $p_{\rm T}$ intervals. A hardening of the
$p_{\rm T}$-differential cross section with the collision energy is observed,
while, for a given energy, $p_{\rm T}$ spectra soften with increasing rapidity
and, conversely, rapidity distributions get slightly narrower at increasing
$p_{\rm T}$. 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. None of the considered models manages to describe the
evolution of the cross section with $p_{\rm T}$ and rapidity at all the
energies.
