Unless you live in a subarctic climate you likely have a poorly insulated house, inefficient heat, or both. My previous house was a 2400 sqft modern home on a heat pump and I was paying ~$80 a month for 24/7 heat at 72 degrees.

That said:

Depending on the circumstances, running the heater “for a few hours in the morning” can be more expensive than keeping the heat on all the time. Let’s say it’s 20 degrees outside at night. A well insulated home can take almost the entire day to cool down from 70 to 20, and then you need to run the heater at full duty cycle for a long time to get it back up to temp for your morning routine.

The energy difference between keeping a house at 70 all day versus heating it from 20 to 70 every morning might be a lot smaller than you expect, so even with gas heating it might not me as much extra as you expect to keep the heat on all the time. Furthermore, if you have a heat pump heater it will kick on low efficiency auxiliary heaters if there is a large difference between the desired temp and the current indoor temp. Under those circumstances it will be WAY more expensive to run the heater for a few hours each day than to keep it on.

Also, usually when we talk about parent fighting with their kids about the thermostat, it’s usually a fight over whether to set it at 65 vs 70, not whether or not you have heat at all. Setting it 5f lower is going to be a much smaller difference than simply not using it.