No heated hostility (4,3)
I believe the answer is:
cold war
'no heated hostility' is the definition.
'cold war' can be an answer for 'hostility' (cold war is a kind of hostility). I'm not certain of the 'no heated' bit.
'heated hostility' is the wordplay.
'heated' becomes 'cold' (I can't justify this - if you can you should give a lot more credence to this answer).
'hostility' becomes 'war' (war is a kind of hostility).
'cold'+'war'='COLD WAR'
Can you help me to learn more?
(Other definitions for cold war that I've seen before include "Low card (anag.)" , "(Freezing?) state of hostility between nations" , "East West tension" , "Conflict without engagement" , "Period of East-West political hostility" .)