A cooler division trophy? (5)
I believe the answer is:
award
'trophy?' is the definition.
(trophy is a kind of award)
'a cooler division' is the wordplay.
'cooler division' becomes 'ward' (I have seen 'Division of borough ' mean 'ward' so perhaps 'division' could also mean 'ward'. I am not sure about the 'cooler' bit.).
'a'+'ward'='AWARD'
Can you help me to learn more?
(Other definitions for award that I've seen before include "medal, perhaps" , "They are disinclined to work" , "Present - prize" , "Bestow" , "Grant made by a court of law" .)