Pets. Other family members. Plus the fact that you're "daytime" heating/cooling a home from 6:30-8:30 am and 3:30-10 pm anyway, and that modern or updated homes keep their temperature for several hours. So the debate is really over a few degrees for a couple of extra hours (Vs. heating an entire other building that may not otherwise exist).
I know at our workplace heating on Monday mornings has to start several hours earlier than on Tuesday because the office lost almost all of its built-up heat from the previous Friday. I also know that they continue to run the heating on holiday-Fridays/Mondays because if they don't by the following Monday/Tuesday, the temperature will drop too low and could damage equipment.
Well, you may live with other people who are in it when you're not, causing it to be heated anyway.
But I guess the answer would be that people can't abide the thought of one uncomfortable minute in their own home, and don't bother figuring out how to program the thermostat.
Typical American house uses a heat pump (basically an AC run "backwards") with a single thermostat (bigger houses might have multiple complete systems).
If you turned the heat off at 7am as you left for work, the house would probably be below 60 when you got home, and take most of the evening to reheat, only to be turned back down again at bed time. I don't know anybody who manually does any of this - at best, they have a smart thermostat that lets them schedule home/away time or uses cell phones as presence sensors. And even then, they'll lower the heat 5-8 degrees, not turn it off completely.
That might be typical for new builds, but it varies a lot by region. Much of the northern half of the country is still burning propane, pellets/wood, or fuel oil for heat.
Yes it follows from first principles, a well-insulated house has a structure that is being heated and retains heat (like a pizza stone) as well as the air itself. Heating air inside a cold structure is quite different from heating air inside a warm structure. So your evening heating will have to work harder to warm the structure back up that was allowed to cool in the day. The air in the evening will cool faster and the heating will have to kick back on again, dumping heat into the structure.
If you leave the heat on during the day (and that can be a few degrees lower) then you can keep the structure warm and avoid all that evening heat loss and cycling. But only if you have good insulation. Otherwise you’re just dumping heat out into the environment. The other factor is how cold it is outside, if the temperature gradient is steep then it’s harder to contain hot air.
You can think about this at the limits. You could turn off the system for days, weeks, months, or whole seasons and allow the house to drift with the ambient environment. Then, you turn it on and recover to the conditioned mode. This certainly uses less energy than if you maintained it for that whole duration. This is because the loss to environment tapers off. It does not keep losing and reach some absurd extreme beyond that of the environment. Reducing the interval for the drift does not fundamentally change this equation. At the other extreme, you can do what a normal thermostat does now. The system is turned off for minutes and allowed to drift until it is called back into action. All other time intervals in between exist on a continuum. The maximal losses are when you try to keep the constant comfort level.
As far as thermodynamics, you're always exchanging heat with the environment. Only the coefficients change with different insulation levels. And these losses correlate with the gradient between the inside and outside. When the gradient is zero, equilibrium means zero net heat transfer. When the house is allowed to drift towards the outside temperature, the total losses will be lower than if comfort were maintained through those same hours. The integral (sum) of these loss rates over time is your total energy loss and a good proxy for the total energy needed from the heating system to condition the space.
The whole tradeoff is about comfort and convenience to have the space conditioned when you want it and to have an appropriately sized system for the needed load. Whether oversized or undersized, equipment may not operate efficiently if asked to operate outside its designed load level and duty cycle. It can also become unreliable with the wrong duty cycle, i.e. a small unit asked to run too long and too frequently or a large unit asked to run too infrequently and for too short a duration each time.
That the equipment has to operate at a higher load during recovery does not imply that it actually uses more total energy in the daily cycle. That would only be true if the equipment is inefficient at the recovery load. An example where this is true would be a heatpump system with an auxiliary resistive heating element that engages in recovery. But, a gas furnace or a sufficiently large heatpump is likely more efficient in recovery since it is also working with a larger gradient until recovery is complete.
It is common for basic clock-based, programmable or smart thermostats to allow for daily drift to save energy. They use a more comfortable set point during morning and evening hours when occupants are most sensitive and then allow some drift towards ambient during midday and nighttime hours when occupants are less likely to notice. This is precisely to leverage this tradeoff to have less total energy losses per day. They don't completely turn off but vary the set point so that the building is kept within a range where the comfortable and efficient recovery is possible. Depending on the day, this might be equivalent to turning off or it might just reduce the duty cycle slightly.
It depends on what you mean by "let it get cold" and how good your insulation is and a number of other factors.
But thermal mass can be a big thing. My house isn't even all that well insulated, and when it is 0º F outside, it only drops about 10 degrees if the furnace is off all day.
