The PHENIX experiment at the Relativistic Heavy Ion Collider has measured the differential cross section of ϕ ( 1020 ) -meson production at forward rapidity in p + p collisions at √ s = 510     GeV via the dimuon decay channel. The partial cross section in the rapidity and p T ranges 1.2 < | y | < 2.2 and 2 < p T < 7     GeV / c is σ ϕ = [ 2.28 ± 0.09 ( stat ) ± 0.14 ( syst ) ± 0.27 ( norm ) ] × 10 − 2     mb . The energy dependence of σ ϕ ( 1.2 < | y | < 2.2 , 2 < p T < 5     GeV / c ) is studied using the PHENIX measurements at √ s = 200 and 510 GeV and the Large Hadron Collider measurements at √ s = 2.76 and 7 TeV. The experimental results are compared to various event generator predictions (pythia6, pythia8, phojet, ampt, epos3, and epos-lhc).