I was typing a reply similar, but you beat me to it.
Without designing/buying something that's specifically tuned to the sound you want to detect, as stated, it will provide false triggering due to other sounds.
You could monitor the voltage on the wire that triggers the heat to turn on/off, or, as already stated, monitor the heat. I did something similar a few years ago to determine how many times my dehumidifier turned on/off.
I took all the temperature data and used Excel to first detect when warmer air was leaving the unit. This told me when it came on, but required I manually counted each turn on. After I took that data and used Excel to determine when the system turned off. Finally, I did a count on each chunk.
It was a bit more involved trying to get Excel to tell me what I wanted, but that's the summarized version.