How common is it to have pain under a temporary crown?

It's possible: It depends on a few factors. Was the tooth symptomatic prior to the temp. Being paced? Was the crown placed due to a deep cavity or crack? Or it is possible the bite on the temp is high. All of these factors could mak discomfort possible, but all require you to tell your dentist so they can help.

Possible but: First is the tooth root canal treated.? If it is then it night be infected or the crown has a high spot causing the pain. If the tooth is vital, then it might be over prepped for the crown, meaning too much enamel was removed to prepare for the crown making the tooth hypersensitive

Check with your dent: If you had a very deep pulp cap or deep decay removal, or you had severe cracked tooth syndrome with symptoms and pain before you had your crown prep done, it is possible to have pain during temp crn phase, but check and make sure you may need root canal or other treatment.

Pain: Pain is a red flag that something's wrong. You just had surgery. May be sore for a few days but pain should rapidly diminish. Tooth may be split. Temporary crown may be too "high." Nerve of tooth may be infected. Many other reasons for soreness, but the symptom should not be ignored. Call your Dentist to report all symptoms and seek his/her advice going forward.