Do outdoor pools have thermostats that are controlled by air temperature?

No. It would be very uncommon for a thermostat to determine whether to turn the pool heater on based on the outdoor AIR temperature. All pool thermostats monitor the WATER temperature. In this way, they keep the water at a predetermined temperature (just like your house thermostat). When the water is cooler than the predetermined setting, the thermostat turns the heater on. When the water is cooler than the preset temperature, the heater is turned off. In this manner the pool water maintains the desired temperature regardless of changes in the air temperature. You could re-wire your thermostat to use air temperature rather than water temperature to turn the pool heater on/off so that whenever the AIR becomes colder than a preset temperature (and for as long as the air temperature stayed below that value), the heater would come on. The potential problem with this arrangement is that the heater could be continually on (as long as the air stayed cooler than the preset value) and the water would be continuously heated to a VERY high temperature! Conversely, if the air were continually warmer than the preset value, the heater might never turn on. Hope this helps ...