Yes, it is normal to have some sensitivity after having a crown done. It is normal anytime a tooth is drilled on to have cold sensitivity. In most crown cases a chord is placed inside the gums for the final impression to be taken, so it is normal for the gums to be sore. Usually it is recommended for the patient to take ibuprofen over the counter or a prescription can be given to help with the post op discomfort for 24-48 hours.