In an intense way (6)
'in an intense way' is the definition.
(I've seen this in another clue)
This is the entire clue.
(Other definitions for hotly that I've seen before include "in an ardent manner" , "Heatedly" , "With zeal - as one may be pursued" , "showing passion" , "disputed thus?" .)
(Other definitions for deeply that I've seen before include "With profundity" , "Thoroughly" , "fathoms down?" , "With intense feeling" , "In a profound way" .)