1.
If people discuss, argue, or say something hotly, they speak in a lively or angry way, because they feel strongly.
The bank hotly denies any wrongdoing.
= vehemently
ADV: ADV with v
2.
If you are being hotly pursued, someone is trying hard to catch you and is close behind you.
He'd snuck out of America hotly pursued by the CIA.
= closely
ADV: ADV with v