French insolence - endlessly cool (6)
I believe the answer is:
gallic
'french' is the definition.
(I've seen this before)
'insolence endlessly cool' is the wordplay.
'insolence' becomes 'gall' ('gall' can be a synonym of 'insolence').
'endlessly' means to remove the last letter.
'cool' becomes 'ice' (icing is a kind of cooling).
'ice' with its final letter taken away is 'ic'.
'gall'+'ic'='GALLIC'
(Other definitions for gallic that I've seen before include "Typically French or of France" , "Pertaining to France" , "Like certain wars" , "Of the French/France" , "Characteristic of the French" .)