The energy dependence of the photon-proton total cross section, σ_tot(γp), was determined from e⁺p scattering data collected with the ZEUS detector at HERA at three values of center-of-mass energy, W, of the γp system in the range 194<W<296 GeV. This is the first determination of the W dependence of σ_tot(γp) from a single experiment at high W. Parameterizing σ_tot(γp) α W^2ε, ε=0.111 ± 0.009 (stat.) ± 0.036 (syst.) was obtained.