PID is a regulator. If you want anything more complex than a constant reference signal for it to regulate to, you're talking a supervisory controller with a programmed feedforward signal. Once you have that, trivial extension to further optimize based on weather reports etc.
properly programmed they should be able to cut heating and cooling bills (and joules) significantly
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That arguably justifies a smart thermostat, but it still doesn't justify an internet connection.
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