The concept of "it takes more power to start the A/C than it takes to keep it running" is a myth, at least for residential users who do not pay separate "Demand Charges" (industrial and commercial users do, but A/C is not likely going to make a difference anyway).
But more importantly, and not related to the ELECTRICAL issue, is how Air Conditioning functions. The fan just moves air around, but that is actually an important feature, because air movement is PERCEIVED by us as being cooler (because air moving over our skin allows our evaporative cooling system, called "sweat", to function. That's why ceiling fans are often used to help us feel better at higher temperatures.
The actual removal of heat by an A/C unit is accomplished by compressing a refrigerant. A room temperature sensor (thermostat) makes the decision on when to cycle the compressor on and off, and does so based on a "hysteresis" model, meaning the upper and lower acceptable limits. So for instance if you set the thermostat for 78F, the hysteresis is likely that it turns on at 80 and turns off at 76, so your average temperature is 78. That compressor is the real energy hog, not the fan. The fan is costing you pennies, the compressor is costing you dollars.
By turning the A/C off on your own, you are simply messing with the hysteresis model based on what you can tolerate / notice. The problem with that is that the models built into an A/C unit are based on its cooling capacity, measured in BTUs (British Thermal Units), meaning it's ability to remove heat. That capacity must be FASTER than the building's ability to ABSORB heat form the outside, referred to as its RESISTANCE to heat transfer, called the "R Value", often seen in describing insulation. But the higher the heat is inside your building, the longer the A/C must run to overcome it, AND THEN keep it cool inside as heat is being absorbed from outside. This is referred to as "thermal mass", meaning EVERYTHING in your house absorbs and distributes heat. So if the house is allowed to get hot, so do the walls, floor, ceiling, furniture, clothes, belongings, appliances etc. etc. When you then turn the A/C back on, it must cool down ALL of those things first before you start to notice the air temperature changing. Leaving the A/C running means it ONLY has to deal with the new heat coming in from outside. In general, it therefore uses less energy.
A secondary aspect, probably important in New York and lots of other places, is that an A/C unit also dehumidifies your air, which is another way we perceive "coolness". So by turning it off yourself, you are allowing the humidity to increase again, you feel hotter sooner, so you end up having to run the A/C longer before you feel comfortable.
Your best way of saving energy is to set the thermostat higher, leave it on all day, and get fans blowing ON YOU that make you feel more comfortable at the higher temperature.
