There is a saying I remember my dad telling me when I was younger: "Those who listen at doors hears nothing good of themselves."

The answer is: No. Don't go through your husband's phone without his permission.

If it turns out he his innocent of any wrong-doing, you've broken serious trust with him and will probably risk losing your relationship.

If he is doing something wrong, you may or may not actually find out, through his phone.

If you're concerned that he is cheating, or doing something wrong, you're much better off seeking help through a counselor or therapist together. It sounds like there is a serious case of unmet needs or trust needs here, and that is a much bigger conversation.