Will it still hurt when I have to get a crown put on my broken tooth?

Not likely. Sensitivity is not uncommon after a crown is placed over the broken tooth. If this goes away, 6-8 weeks, then tooth is recovering and getting back to normal. However, the transition from temperature sensitivity to constant pain is a bad sign. There is a chance you are going to need an endodontic treatment. See your dentist for re-evaluation if needed. Good luck.
Talk to dentist. A crown will go a long way to making your tooth strong and healthy again and stopping the pain you are having. You dentist will determine the cause for the pain before the crown is done and if you will need a root canal or another treatment first.
No. If you have had no treatment yet and your tooth hurts now, a crown alone may not be the solution. You may require root canal treatment first. If your broken tooth is just sensitive now, a crown will likely help. If you've already had a crown started and you have pain, you may need root canal treatment. Usually placing a new crown on a healthy tooth doesn't cause pain.