Everyone, in every language, refers to temperature, probably multiple times a day, whether referring to themselves, the weather (particularly the British!), or without necessarily realising it or any presence of noticeable physical temperature (read on for this last one). Let’s begin with the basic words hot and heat; the difference between the adjective and the noun is merely a change of vowels. This alteration of an internal vowel is known as an ablaut inflection. It is a very ancient trait of Indo-European languages which has all but disappeared from most modern languages although it is still quite common in English. Thus we have sit, sat and seat, raise, rise and rose, draw and drew and many others. The entire process seems so natural to speakers of English that Americans invented the word dove (as in swimming pool, not as in aviary) as the past tense of dive. (The British word is dived.)
Another ablaut inflection is Het, being the past tense of the verb to heat. So, to get all het up is to become “all heated up”. A heat meant the same as a heating. Thus a cookery book of 1430 instructs us to “sette it on the fire an giffe it an hete” where we might have said “set it on the fire and give it a heating”. This sense of heat led to its use in racing. Before racing a horse it would be given a preparatory run. Today, we would refer to this as a “warming-up exercise” but in the 16th century it was called giving the horse a heat (i.e. “a heating”). In time, any preliminary run before a final race became known as a heat and when two runners cross the finish line together it is called a dead heat.