Hotly means with strong and angry feeling. Hotly denied, hotly contested, argue hotly.
It fixes itself to a few verbs and refuses others — hotly denied and hotly disputed are everywhere, hotly agreed is not. The heat is the metaphor English has used for anger for a thousand years. Note the Old English sense, meaning fervently.
1adverbhotlyWith strong and angry feeling.
“The two teams hotly disputed the referee's decision.”
hotly is a manner of feeling.
Old English hatlice means ardently or fervently, from hat, hot, plus the -lice ending that makes adverbs.
Old English
hātlīċe — ardently, fervently
Middle English
*hotly
English, today
hotly
From Middle English hotly, hatliche, from Old English hātlīċe (“ardently, fervently”); equivalent to hot + ly.
Etymology from Wiktionary, CC BY-SA