Critical thickness and strain relaxation of c-plane GaN layers grown by molecular beam epitaxy on AlN were studied as a function of growth temperature and threading dislocation density (TDD). For this purpose we used AlN/sapphire templates and AlN single crystals with TDDs of similar to 10(9) cm(-2) and similar to 10(3) cm(-2), respectively. Whereas at high growth temperature (900 degrees C) the critical thickness for plastic relaxation is only 3 monolayers (MLs) for both substrates, this value drastically increases when decreasing the growth temperature. It reaches similar to 30 MLs when GaN is deposited at 750 degrees C on AlN single crystals. We also observed that the strain relaxation rate strongly depends on TDD. These results exemplify the lack of efficient gliding planes in III-nitrides when grown along the c-axis, which, combined with low kinetics, allows for plastic relaxation to be frozen out. Achieving pseudomorphic GaN layers on AlN is of interest for two-dimensional electron gases based on AlN/GaN/AlN heterostructures lattice-matched to AlN single crystal substrates.