old vs past times

old and past times both are nouns.

Word	Noun	Adjective	Verb	Adverb
old	Yes	Yes	No	No
past times	Yes	No	No	No

As nouns, past times is a hypernym of old; that is, past times is a word with a broader meaning than old:

Other hypernyms of old include past, yesteryear.

old (noun)	past times (noun)
past times	the time that has elapsed

old (adjective)	past times (adjective)
of long duration; not new
(used especially of persons) having lived for a relatively long time or attained a specific age
skilled through long experience
(used for emphasis) very familiar
excellent
just preceding something else in time or order
belonging to some prior time