Sensitivity if often due to a nerve being exposed due to decay or the cracking of a tooth, or lost filling. This most often results is a root canal.
many times there is residual sensitivity due to mild trauma while drilling. if it doesnt feel better soon a repeat visit is advised but with the only symptom being mild cold sensitivity nothing indicates an impending root canal.
You shouldn't have any sensitivity to cold or heat. It sounds as if maybe there is an opening somewhere in the filling or the tooth is cracked. If it continues I would go back to him, you may also need a root canal which your dentist should have known.
since you are describing it as discomfort instead of actual pain i would give it a bit more time. perhaps try a toothpaste for sensitive teeth. watch out for heat sensitivity which could mean you have an infection and if the cold sensitivity remains after a week or two contact your dentist. the drilling can irritate the nerve and a calming down period could be necessary.