Hot and cold compress useful for trigeminal neuralgia(TN)

Condition analysis:
It could be, but would be atypical and not very likely.  The first branch of the trigeminal nerve innervates the forehead and surface of the eye. The second branch cover the upper teeth, the upper lip, part of the nose and part of the cheek.  The third branch is the lower jaw, lower teeth and part of the tongue.

There is a type of headache called a "Cluster headache" that is typically described as being "behind the eye."  Cluster Headaches are called Suicide Headaches due to their pain factor.

Yours are far too short in duration, but there is a similar condition called: Chronic Paroxysmal Hemicrania (CPH) is a condition similar to cluster headache, but CPH responds well to treatment with the anti-inflammatory drug Indocine and the attacks are much shorter, often lasting only seconds.

Instructions:
Hot And Cold Compress: When the nerve begins to send shooting sensations of pain, fill an ice bag with ice cubes and place it on the side of the face where the pain is. Keeping the ice bag in place, heat some water in a basin and dip a soft towel into it. Wring out the excess water and fold it several time to form a compress. Remove the ice bag and place the hot compress on the face. Do this therapy, alternating the hot and cold compresses for fifteen minutes. The pain will immediately subside and bring tremendous relief.