I derive the mass-temperature relation and its time evolution for clusters of galaxies in different cosmologies by means of two different models. The first one is a modification and improvement of a model by Del Popolo & Gambera, namely based upon a modification of the top-hat model in order to take account of angular momentum acquisition by protostructures and of an external pressure term in the virial theorem. The second one is based on the merging-halo formalism of Lacey & Cole, accounting for the fact that massive clusters accrete matter quasi-continuously, and is an improvement of a model proposed by Voit, again to take account of angular momentum acquisition by protostructures. The final result is that, in both models, the M-T relation shows a break at T~ 3-4 keV. The behaviour of the M-T relation is as usual, M~T3/2, at the high-mass end, and M~Tγ, with a value of depending on the chosen cosmology. Larger values of γ are related to open cosmologies, while Λ-cold dark matter (ΛCDM) cosmologies give results of the slope intermediate between the flat case and the open case. The evolution of the M-T relation, for a given Mvir, is more modest both in flat and open universes in comparison with previous estimates found in the literature, even more modest than what found by Voit. Moreover, the time evolution is more rapid in models with L= 0 than in models in which the angular momentum acquisition by protostructures is taken into account (L≠ 0). The effect of a non-zero cosmological constant is that of slightly increasing the evolution of the M-T relation with respect to open models with L≠ 0. The evolution is more rapid for larger values (in absolute value) of the spectral index, n. The mass-temperature relation, obtained using the quoted models, is also compared with the data by Finoguenov, Reiprich & Böeringer. The comparison shows that these data are able to rule out very low-Ω0 models (<0.3), particularly in the open case, and that a better fit is obtained by ΛCDM models and by CDM models with Ω0 > 0.3.