The photoproduction of $D^{*pm}$ mesons has been measured with the ZEUS detector at HERA at three different ep centre-of-mass energies, $sqrt{s}$, of 318, 251 and 225 GeV. For each data set, $D^*$ mesons were required to have transverse momentum, $p_T^{D^*}$, and pseudorapidity, $eta^{D^*}$, in the ranges $1.9 < p_T^{D^*} < 20$ GeV and $|eta^{D^*}|<1.6$. The events were required to have a virtuality of the incoming photon, $Q^2$, of less than 1 GeV$^2$. The dependence on $sqrt{s}$ was studied by normalising to the high-statistics measurement at $sqrt{s} =318$ GeV. This led to the cancellation of a number of systematic effects both in data and theory. Predictions from next-to-leading-order QCD describe the $sqrt{s}$ dependence of the data well.