The acquisition of mammalian endothermy is poorly constrained both phylogenetically and temporally. Here, we inferred the resting metabolic rates (RMRs) and the thermometabolic regimes (endothermy or ectothermy) of a sample of eight extinct synapsids using palaeohistology, phylogenetic eigenvector maps (PEMs), and a sample of 17 extant tetrapods of known RMR (quantified using respirometry). We inferred high RMR values and an endothermic metabolism for the anomodonts (Lystrosaurus sp., Oudenodon bainii) and low RMR values and an ectothermic metabolism for Clepsydrops collettii, Dimetrodon sp., Edaphosaurus boanerges, Mycterosaurus sp., Ophiacodon uniformis and Sphenacodon sp. A maximum-likelihood ancestral states reconstruction of RMRs performed using the values inferred for extinct synapsids, and the values measured using respirometry in extant tetrapods, shows that the nodes Anomodontia and Mammalia were primitively endotherms. Finally, we performed a parsimony optimization of the presence of endothermy using the results obtained in the present study and those obtained in previous studies that used PEMs. For this, we assigned to each extinct taxon a thermometabolic regime (ectothermy or endothermy) depending on whether the inferred values were significantly higher, lower or not significantly different from the RMR value separating ectotherms from endotherms (1.5 ml O₂ h^-1 g^-0.67). According to this optimization, endothermy arose independently in Archosauromorpha, Sauropterygia and Therapsida. This article is part of the theme issue 'Vertebrate palaeophysiology'.