I got a deep cavity filled three months ago - recently it has developed extreme sensitivity to cold food, liquids and air. What could it be?

Possible root canal. Sometimes fillings can lead to sensitivity especially if it is deep. However, if the pain is getting worse, keeping you up at night, or you have lingering pain with cold, it may be a sign you need a root canal treatment. Please see you dentist for an evaluation.
Pulpal hyperemia. Also called a reversible pulpitis. When caries and bacteria are in proximity to pulp, toxins and even bacteria may enter pulp resulting in inflammation. Cold/hot causes pain for less than 10sec., it is a reversible condition. When the pulp is severely inflamed and spontaneous pain or lingering pain to cold/hot stimuli (>20 sec) it is irreversible pulpitis. Rec. See a dentist for diagnosis.