Can fat freezing cause damage to the baby in the early stages of pregnancy?

Thank you for your question and congratulations on your pregnancy! As far as I am aware of, fat freezing should not have negatively affected your pregnancy. 

While I would not recommend CoolSculpting to anyone who is already pregnant — simply because I never recommend unnecessary procedures to pregnant women — you shouldn’t worry. CoolSculpting’s patented cryolipolysis technique, which freezes the fat cells and sends them into a slow death cycle, is only low enough to damage fat cells. It does not affect surrounding tissues or organs. Congratulations on your pregnancy! Be sure to consult an OB/GYN right away. Mention the CoolSculpting to them, but don’t worry about it. After you’ve delivered a healthy baby and lost the baby weight, a board-certified dermatologist could help you with your next CoolSculpting session, if necessary. 

CoolSculpting freezes fat cells and causes them to undergo apoptosis.  The dead fat cells are then cleared by your lymph system.  Your lymph system is always busy clearing cells, so there should be no danger.  Also, there is a good chance that you only had placental tissue and amniotic fluid and the fetus had not even started developing by that time.
That having been said, I would not recommend CoolSculpting during a known pregnancy as there is not data that I know about.  Also, your body changes with pregnancy and breast feeding so waiting 6-12 months from the end of nursing for CoolSculpting is a good idea.