(Edit - -22nd Jan) Please note. I have spotted an error in this post and I will be correcting it in a new post.
Its an interesting question, is it more economic to run a fridge on gas or electric.
Obviously it depends on how much you pay for either gas or electric, so to be able to make a reasonable comparison you need to look at the quantity of each fuel used, and then you can work out how much either fuel will cost you.
The easiest common denominator to use is cost per kWh.
With electric power its easy because most appliances already quote their eclectic power consumption in Watts or Kilowatts.
By comparison gas consumption is usually quoted in grams per hour. LPG Butane and Propane typically have Butane 13.6kWh/Kg Propane 13.8kWh/Kg for this purpose well use 13.8kWh/kg
Here are the specifications for a Dometic RMS 10.5T fridge.
Voltage (AC) 230 V
Rated input power (AC) 135 W
Gas consumption [g/24h] 270.00 g/24h
Gas pressure (DIN732) 30 mbar
Based on the conversion factor 270g/h x 13.8= 369.9 W/h. This model therefore uses at least 2.73 times as much energy on gas as it does on electricity.
But this is not the final comparison, as when on eclectic power the thermostat will cut the power completely, whist on gas the fridge continues to use gas to maintain the pilot light which typically is roughly about 100W. It is therefore almost certainly the case that fridges will typically use 3x more gas watts than if it ran on electric.
On that basis, in most cases it will almost certainly be cheaper to run the caravan fridge on mains electric than gas. (Edit added later) Its likely that fact will remain much the same as the cost of energy how ever its provided seems to retain similar margins over time.
hen there is the convenience factor, It is reasonable to assume that if you have chosen a pitch where 230V ac is available, you are likely paying more for the pitch. so why not use it. By doing so you are also not using as much gas, which leaves more for jobs where electric is insufficient such as cooking, space or water heating. You'll change your bottle less frequently, saving the cost and trouble of having to take it to be exchanged or refilled.
However, the cost argument reverses if you use a portable generator to make electricity. LPG is far more cost effective than petrol through a generator.