It depends... either propane or D/C
I keep my r-pod in a storage lot and generally don't go out the day ahead to cool the fridge down. Instead, on the day I head out, I'll head to the lot, turn on the propane and load up the fridge with food. With a full fridge of already cold food/beverages, the propane works very quickly. I've never had any thawing issues - even with icecream/froyo.
On a trip of multiple stages, I'll usually put the fridge on D/C while travelling between campgrounds. The battery starts out fully charged, and the charge line from my vehicle provides sufficient power for the fridge and to keep the battery charged up. No need to waste propane (although, admittedly, the fridge uses very little propane).
When I've been boondocking and head back home, I'll put the fridge on propane. Generally, the battery will be a bit low when I leave the camping area and I'll want to fully charge it from the vehicle while towing home as the r-pod is headed straight for the storage lot (which doesn't have electric). If the fridge is on D/C, a couple hours of driving won't provide enough power to also charge the battery.
In answer to the unspoken question... no, I don't worry about driving with propane on. I can understand why others hesitate to do so, but I don't share their concerns. Of course, I do shut the propane off before filling my vehicle with gasoline and tunnels and the odd bridge also have restrictions where I'd shut off the propane.