Is it okay if my boyfriend checks in on his ex from time to time?

My boyfriend of one month left his girlfriend for me basically. I met him the beginning of February we started off as friends and I’m guessing he started to like me so the week after Valentine’s Day he broke up with her and told her that he had met me. For Valentine’s Day he bought her a bag full of bath nd body works, shoes, card and took her out. But from what he told me is that he left here because her family didn’t like him and she basically let them control her. He also told me that they worked together and that’s how they met. Other than that he said she was a sweet girl and she loved hard. Yesterday my boyfriend let me use his phone while we were out and i saw that he had texted his ex the night before at 8:30. His text said “hey how are you doing” (she didn’t text back) as i went through their text log and started reading i could tell this girl loved him just by her words. The day after we made our relationship official he texted her and asked her was she at work that day (I’m guessing he wanted to see if she was at work so that he could see her... idk) i also saw text messages from days where we were going out and she had texted him telling him she missed him and he said he missed her too and the Friday we were supposed to see each other she asked to see him and they were making plans to hang out with each other. But ever since we have been a couple their texts have died down so I’m guessing he’s starting to distance himself from her.
Is it okay if my boyfriend checks in on his ex from time to time?
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