SO calling another girl by pet names?

Crys

UPDATE AGAIN: So I told him I went through his messages with even though he had told me he would leave if I ever went through his phone but I couldn't pretend I wasn't hurt and angry. I was expecting him to just make it all about his phone and how he wasn't at fault because I am the one that went through his phone. He surprised me though and didn't even argue with me. He admitted that what he did was wrong and that he was so sorry. He doesn't want to lose me and he wants to work on our relationship because he wants his family. He asked if having an open phone policy would help and I agreed. He messaged her and told her that he will not be talking to her anymore because he crossed a line as well as she did. I am still upset and I don't know when I will get over it but do you guys think that this is good? I really don't want to leave him... But is there anything else I should be doing? Any tips or advice would be helpful if you have gone through something similar.

UPDATE: SO I TOOK SOME SCREEN SHOTS OF THE TEXTS THEY HAVE BEEN SENDING BACK AND FORTH. SOMEONE TOLD ME I WAS OVER REACTING SO THE SUPPORT PROVING I AM NOT WOULD BE GREAT BUT IF I AM PLEASE LET ME KNOW! THE CONVERSATIONS BETWEEN HER AND HIM HONESTLY REMIND ME SO MUCH OF WHEN WE STARTED SEEING EACH OTHER 3 YEARS AGO.

So maybe I am over reacting but I am curious. Would any of you be upset if your SO was calling another girl "cutie", "lovely" and "beautiful" in text conversations. For example he will message her in the morning saying "Good morning Cutie!" and such. She also sends him pictures (nothing inappropriate) and he comments on how cute she is and vice versa. We have been together for 2 years and I am currently 8 months pregnant. A part of me is worried if it's just my hormones but the other part of me thinks it isn't right, especially since he never says anything like that too me. The last time he did was when we first started out dating...