We report the effect of surface temperature on the state resolved translational energy distributions for H-2 and D-2 recombinatively desorbed from Cu(111). Sticking functions S(upsilon,J,E) can be obtained by applying detailed balance arguments and follow the familiar error function form at high energy, consistent with previous permeation measurements [Rettner et al., J. Chem. Phys. 102, 4625 (1995)]. The widths of the sticking functions are identical for both isotopes and are independent of rotational state. S(E) broadens rapidly with increasing surface temperature, with a low energy component which is slightly larger than represented by an error function form. This is similar to the behavior seen on Ag(111) [Murphy et al., Phys. Rev. Lett. 78, 4458 (1997)] but on Cu(111) the low energy component remains a minor desorption channel. The broadening of S(E) can be explained in terms of a change in the distribution of barriers caused by local thermal displacement of the surface atoms, thermal activation of the surface producing sites where molecules can dissociate, or desorb, with a reduced translational activation barrier. At low energy sticking increases rapidly with surface temperature, with an activation energy of 0.54 and 0.60 eV for H-2 and D-2, respectively. These values are similar to the thermal activation energies calculated for translational excitation of H-2/D-2 and imply that thermal excitation of the surface is just as efficient as translational energy in promoting dissociation. The influence of surface temperature decreases with increasing translational energy as molecules become able to dissociate even on the static Cu(111) surface. By comparing the energy distributions for desorption with existing angular distributions we determine how the effective energy, E-e = E cos(n(E)) theta which contributes to adsorption-desorption, scales with translational energy. At translational energies near the threshold for sticking n(E) approximate to 2, sticking scales with the normal component of the translational energy and is not influenced by motion parallel to the surface. At lower energy n(E) drops towards zero, indicating that motion parallel to the surface aids dissociation, consistent with dissociation at a corrugated barrier.