Fast scanning calorimetry has been applied in order to understand the phase transitions in thermoplastic elastomers (TPEs) based on well-defined multi-block copolymers made of 'soft' polytetrahydrofuran and 'hard' terephthalate ester diamides. The intrinsically complex chemical structure of TPEs leads to complex phase transitions. By changing their thermal history over a wide range of temperature (from -100 degrees C to 200 degrees C) and cooling rates (from 10 to 4000 degrees C s(-1)), we clarify the origins of the various phases present in these materials. In particular, we study the different possibilities for the hard segments to associate depending on their mobility during the quenching phase, forming either strong and stable structures or weaker and metastable ones. Besides, we demonstrate that a minimal cooling rate of 800 degrees C s(-1) is necessary to keep these TPEs (made of short and monodisperse hard segments) amorphous leading to a subsequent cold crystallization when heating back, at around 30 degrees C. Finally, we validate our interpretations by varying the copolymer composition (from 10 wt% to 20 wt% hard segments), revealing the thermal invariance of poorly organized domains. Based on these data, we also discuss the importance of chain diffusion in the crystallization process. Applying fast scanning calorimetry allows us to link fundamental understanding to industrial application. (c) 2018 Society of Chemical Industry