I totally understand not wanting to spend the money to heat the pool in the winter if you are not interested in actually using the pool. I am not sure what the cost will be as of yet, but I plan to keep it heated and ready to use! I put in a pool to utilize it rather than simply look at it. I may change my mind if the cost outweighs the benefit!
Some rough numbers -
Tucson NG costs around $2.50 per therm. A 400,000BTU/hr gas heater uses 400 cubic feet of gas. or 4 CCF (centacubic feet). 4 CCF of natural gas is roughly 4.15 therms (depends on the temperature and specific density of the gas but it's pretty close). So every hour that you run a 400kBTU gas heater you spend a little over $10.
1 BTU raises the temperature of 1 pound of water 1 degree Fahrenheit
11,600 gallons of water weighs approximately 96,744 pounds.
So, to raise the temperature of your pool 1 deg F, requires roughly 97,000 BTUs. A 400kBTU gas heater is only around 80% efficient so your heater delivers about 360,000BTUs/hr to the water. So it will take roughly 16mins for the heater to raise the temp 1 deg F (I suspect a bit longer with conduction and evaporation losses). Based on the gas consumption rate, that's going to cost you $2.69.
So getting a pool from "cold" to swimmable might cost you $50 in gas or more. However, to maintain the temperature the gas heater, and pool pump, have to run all the time. And during the inverted months when the dew point temperature and air temperatures are lower than the pool water temp, you will get significant heat loss from evaporation. That heat loss can easily be 6-8 F overnight during the warm weather months (because the RH is so low) and double that during the winter months.
If you want to maintain a swimmable pool all year round, then you absolutely will need to cover the pool with a bubble cover, assuming the pool doesn't have an automatic cover. You'll probably see gas costs from the pool alone running at $40-$80 per month during the cold weather (Nov-April) and then almost zero cost over the summer when the sun and heat kick in.
And, as a given, with a heater running all the time like that, especially when its cold and flue gas can accidentally condense, you'll be lucky to get 4-5 years of service life from the heater before its shot. So figure every 5 years or so you'll be replacing that heater to the tune of ~$5,000.
Those are just spitball numbers and for some, they're not a problem. I've got other more pressing things to spend m y money on so a swimmable pool is not a priority for me. YMMV.