Utah KJ
Free State of Florida
I also have the ARB 37qt fridge. It has a low voltage protection circuit, that is settable for various cutoff voltages. The long term problem is that the voltage will go low, it will do as designed and cutoff, but you will lose cooling in the fridge. the second problem is that your primary start battery is probably not a deep cycle, so if you kill it more than a few times (even below 11.4 volts) it won't live a very long life.
I wouldn't recommend doing it on a single battery system, unless that single battery was a deep-cycle type battery (like yellow top, or exide, etc). It's asking for trouble.
That said, I run a multiple battery system on my avalanche - 2 under the hood (start battery and aux, separated with a solenoid) and another group 31 deep cycle in the rear cargo area. The aux battery and the group 31 are parallel and get charged from a secondary alternator. The factory alternator, start battery and factory systems are separated unless I manually "boost" the solenoid, at which point the aux is temporarily paralleld to start the engine if the main start is killed. (which for this application is very rare - only factory systems run from it).
I admit It's overkill for just a fridge, but I run a high-output inverter (~3kW) for a work application and threw the fridge in so I'd always have a cold drink, and use for camping!
Or, you can install the kit I posted. Battery protection and cold food win/win