I don't understand the emboldening from CNTRL. How does « chaudement » semantically appertain to « promptement »?
Du lat. pop. tostum, neutre prés. adv. de tostus « grillé, rôti, brûlé », part. passé de torrere « griller, rôtir »; tostum a dû signifier d'abord « chaudement » d'où « promptement ».
Word Origins (2005 2e) by John Ayto. p 510 Left column.
Toast comes via Old French toster ‘roast, grill’ from Vulgar Latin *tostāre, a derivative of the past participle of Latin torrēre ‘parch’ (source of English torrid). Its use as a noun, meaning ‘toasted bread’, dates from the 15th century. It was common to put sippets or croutons of spiced toast into drinks to improve their flavour, and it was the custom of gallants in the 17th century, when (as they frequently did) they drank the health of ladies, to say that the name of the lady in question enhanced the flavour of their drink better than any toast. That is supposedly the origin of the use of the term toast for ‘drinking someone’s health’.