Husband wants to go through my work email. He won't stop. What should I do? And break into my phone. He threatens divorce when I don't.?

Rule 1:- Never give in to threats. 

No matter what you CHOOSE to do the moment threats are issued you need to stand your ground.  Otherwise the threats become ongoing for ever more trivial issues.  You password protect things to prevent him getting access to them.

He doesn't work for your company so absolutely NOT - he should not have access to your work e-mails.  You can and WILL lose your job if your employer finds out.  Boot that abusive controlling POS out to the streets and divorce him!  YOU file for divorce and that will shut him - AND his abusive threats up.  what are you thinking marrying garbage like that?

Let him see your phone.  Problem solved.

He doesn’t trust you. What did you do to give him that impression. You’re leaving something out. You cheat on him?

The husband sounds like a real treat. 

Agree to the divorce. 

He threatens divorce when you don't? Don't WHAT?

He jealous. Either you have given him reason to be jealous or he's crazy. If you haven't done anything then what's the problem. If you have then you shouldn't be married or maybe you could try work on your marriage

That's just it threatening divorce means they're full of hot air! It's time you regained your confident carefree loving ways and kick this loooooooooooooooooooooooser 2 the kerb and find your true soulmate and kindred spirit as this vile specimen is no way that person!

Tell him that you are not allowed to share your work email, that the company told you it's private and you are not to share it and if he wants a divorce, then go ahead and file.  The judge will be on your side when he hears it's over something so petty, especially when he is so controlling.  It will not be tolerated in court.