Hi. You can't be "in a" cross temper. You can be cross (upset/angry) and you can be "in a temper". And you can have a bad temper (which means you get angry very often). Hope it helps.

To say cross temper would mean that a person has a bad temper, but not violent or angry, but rude. And you would 'have' a bad temper, and someone 'is' cross tempered.