Some years ago, the resolved temperature profile of X-ray clusters, observed by ASCA, led us to conclude that there is a break (bent) in the M-T relation. The quoted bent was interpreted in different ways: in some papers it was connected to the effect of the formation redshift,(20) or to cooling processes(29) and heating.(5) In the present paper, we re-derive the mass-temperature relation and its time evolution for clusters of galaxies in different cosmologies by means of a model based on the merging-halo formalism of Lacey and Cole (Mon. Not. R. Astron. Soc. 262, 627 (1993)) modified to take account of angular momentum acquisition by protostructures, of an external pressure term in the virial theorem, and a non-zero cosmological constant. In contrast to previous results, we show that the non-similarity in the M-T relation can be naturally explained in the quoted model that takes into account the amount of the angular momentum of a protostructure. This result is in disagreement with the widely accepted idea that the above non-similarity is due to non-gravitating processes such as those of heating/cooling. The main result of the present paper is that the self-similarity of the M-T relation is broken at the low mas end (T <= 3 keV), and that the evolution of the M-T relation, for a given M-vir, is more modest both in flat and open universes in comparison to previous estimate found in literature, even more modest than that found by Voit (Astrophys. J. 543, 113 (2000)). 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 not equal 0).