  The ratio of the cross sections for inclusive isolated-photon production in
$pp$ collisions at centre-of-mass energies of 13 and 8 TeV is measured using
the ATLAS detector at the LHC. The integrated luminosities of the 13 TeV and 8
TeV datasets are 3.2 fb$^{-1}$ and 20.2 fb$^{-1}$, respectively. The ratio is
measured as a function of the photon transverse energy in different regions of
the photon pseudorapidity. The predictions from next-to-leading-order
perturbative QCD calculations are compared with the measured ratio. The
experimental systematic uncertainties as well as the uncertainties affecting
the predictions are evaluated taking into account the correlations between the
two centre-of-mass energies, resulting in a reduction of up to a factor of
$2.5$ ($5$) in the experimental (theoretical) systematic uncertainties. The
predictions based on several parameterisations of the proton parton
distribution functions agree with the data within the reduced experimental and
theoretical uncertainties. In addition, this ratio to that of the fiducial
cross sections for $Z$ boson production at 13 and 8 TeV using the decay
channels $Z \rightarrow e^+e^-$ and $Z \rightarrow \mu^+\mu^-$ is made and
compared with the theoretical predictions. In this double ratio, a further
reduction of the experimental uncertainty is obtained because the uncertainties
arising from the luminosity measurement cancel out. The predictions describe
the measurements of the double ratio within the theoretical and experimental
uncertainties.
