Hmmmm ... not sure on this one ... according to my refrigerator owner manual, it looks like mine has an instantaneous maximum draw of 6 amps (at 12 volts) and an average consumption over 24 hours of about 400Watt-hours. That calcs to an nominal current draw over a 24 hour period of around 1.5-2 amps. Given that my (upgraded) house battery (a Northstar AGM battery) has a 100amp-hour discharge capacity, if I limit my discharge to 50% and pull no other power, my battery should be able to handle my refrigerator for 24 hours.
I also note that the refrigerator monitors battery voltage coming to it, and shuts down when the voltage is too low (both to protect the compressor and to keep from draining the battery to zero.
If you're only getting 4-6 hours from your house battery before your refrigerator drops off, you might want to check both your house battery's charge state and how many other loads you have running when you are at hook. Since you have a new battery, it shouldn't be sulfated yet (hence able to hold a full charge), but remember that battery life is a function of their average level of charge and you can kill a new battery fairly quickly if you don't bring it back to full charge regularly. Take a look at your battery charger and the "full up" charge level of the battery to see if you are actually charging your battery to 100%.
Incidentally, after talking to Northstar tech reps, we concluded that the OEM Charles Charger on the boat wasn't able to adequately bring/keep their AGM battery at proper charge levels, so I changed out my charger as a part of upgrading my batteries to AGM type.