Sorry for the misunderstanding. You are actually asking a very complicated question. "House" or "Auxiliary" batteries that you are asking about are not typically spec'ed because it is entirely dependent on how you are going to use them and what performance you require.
But lets back up a second. Just to make sure we are on the same page, the "house" battery will run your lights and fans and some other 12V devices, but will NOT run your air conditioner or your microwave. It will run a TV only if you have an inverter installed. (unless you have a 12V TV, which is rare).
Ok, so for what battery you need, we need additional specific information about how you will be using your camper. But I don't think that's the conversation you are looking for.
So, my answer for you is: whatever fits. Take measurements and buy the battery that fits in your box.
A Deep cycle is a better option, but is it required? No. I personally am not using a deep cycle on my Grayhawk.
Hope this helps,