Can I demand for a higher pay?

This is tough as you have already set your rate with your client. I think you can tactfully inform them that you will have to increase your rate as per industry standard --BUT, you must face the risk of getting project termination. In the end, you have to stick to your guns and sell your skills for the rate that you truly deserve. Perhaps, your client won't mind paying you fair if you have been delivering quality work for those time. Good Luck!