No its not normal and to top that off your SES light wont go off if you have a bad thermostat. I had a bad thermostat and i had to diagnose it, clear it, diagnose it again and clear it again before i noticed i had a bad thermostat. the part is around $60 USD.

If i were you id go to a local mom and pop shop & get it diagnosed. If you get codes on the thermostat, replace it.

Good luck!