This paper presents the measurements of π ^±, K ^±, p and p ¯ transverse momentum (p _T) spectra as a function of charged-particle multiplicity density in proton–proton (pp) collisions at s=13TeV with the ALICE detector at the LHC. Such study allows us to isolate the center-of-mass energy dependence of light-flavour particle production. The measurements reported here cover a p _T range from 0.1 to 20 GeV / c and are done in the rapidity interval | y| < 0.5. The p _T-differential particle ratios exhibit an evolution with multiplicity, similar to that observed in pp collisions at s=7TeV, which is qualitatively described by some of the hydrodynamical and pQCD-inspired models discussed in this paper. Furthermore, the p _T-integrated hadron-to-pion yield ratios measured in pp collisions at two different center-of-mass energies are consistent when compared at similar multiplicities. This also extends to strange and multi-strange hadrons, suggesting that, at LHC energies, particle hadrochemistry scales with particle multiplicity the same way under different collision energies and colliding systems.