hunting vs outdoor sport

hunting and outdoor sport both are nouns.

Word	Noun	Adjective	Verb	Adverb
hunting	Yes	No	No	No
outdoor sport	Yes	No	No	No

As nouns, outdoor sport is a hypernym of hunting; that is, outdoor sport is a word with a broader meaning than hunting:

hunting: the pursuit and killing or capture of wild animals regarded as a sport
outdoor sport: a sport that is played outdoors

Other hypernyms of hunting include field sport, blood sport.

hunting (noun)	outdoor sport (noun)
the work of finding and killing or capturing animals for food or pelts	a sport that is played outdoors
the pursuit and killing or capture of wild animals regarded as a sport
the activity of looking thoroughly in order to find something or someone