the.com/hot pursuit

means A chase happening right now, hard on the heels of a fleeing suspect, urgent enough to justify breaking normal rules to catch them.

from A straightforward pairing of 'hot,' used since Middle English to mean fresh, close, or urgent (as in 'hot on the trail'), with 'pursuit,' from Old French poursuite, 'the act of chasing.' The phrase hardened into a legal term of art — especially in doctrines letting police cross jurisdictions or enter property without a warrant when actively chasing a suspect. No hidden story here: the heat is metaphorical, borrowed from the literal warmth of a trail not yet gone cold.