  Simulations of the formation of galaxies, as well as ionisation models used
to interpret observations of quasar absorption lines, generally either assume
ionisation equilibrium or ignore the presence of the extra-galactic background
(EGB) radiation. We introduce a method to compute the non-equilibrium
ionisation and cooling of diffuse gas exposed to the EGB. Our method iterates
the ionisation states of the 11 elements that dominate the cooling and uses
tabulated ion-by-ion cooling and photo-heating efficiencies to update the
temperature of the gas. Our reaction network includes radiative and di-electric
recombination, collisional ionisation, photo-ionisation, Auger ionisation, and
charge transfer. We verify that our method reproduces published results for
collisional equilibrium, collisional non-equilibrium, and photo-ionised
equilibrium. Non-equilibrium effects can become very important in cooling gas,
particularly below 10^6 K. Photo-ionisation and non-equilibrium effects both
tend to boost the degree of ionisation and to reduce cooling efficiencies. The
effect of the EGB is larger for lower densities. Hence, photo-ionisation
affects cooling more under isochoric than under isobaric conditions.
Non-equilibrium effects are smaller in the presence of the EGB and are thus
overestimated when using collisional-only processes. The inclusion of the EGB
alters the observational diagnostics of diffuse, metal-enriched gas (e.g. metal
absorption lines) even more significantly than the cooling efficiencies. We
argue that the cooling efficiency should be considered if ionisation models are
used to infer physical conditions from observed line ratios, as the a priori
probability of observing gas is lower if its cooling time is shorter. We
provide on-line tables of ionisation fractions and cooling efficiencies for
equilibrium and non-equilibrium scenarios. (abridged)
