Coming out for air, regret it's cooler (12)
I believe the answer is:
refrigerator
'cooler' is the definition.
(I've seen this before)
'coming out for air regret' is the wordplay.
'coming out' indicates an anagram.
'for'+'air'+'regret'='forairregret'
'forairregret' anagrammed gives 'REFRIGERATOR'.
'it's' is the link.
(Other definitions for refrigerator that I've seen before include "Kitchen appliance" , "Icebox" , "Cooler" , "Chamber maintaining low internal temperature" , "it will freeze!" .)