Regarding the idea of turning the HP on and off to try to save money... would you plug in and unplug your refrigerator to do the same? Of course not. So the whole plugging/unplugging thing is working really hard to get the HP to operate outside its designed efficient zone.
OP, I suggest an HVAC service call. Have the technician check refrigerant levels and clean both the inside and outside coils. Other than fan obstructions or condensation line clogs, that's about all that could go wrong with a HP.
OP here. Somehow this thread keeps getting derailed from the primary focus, which was improving efficiency, not rants about how much heat pumps suck. Thats not focused on you, ChpBstrd, but from your comment it seems you think I fall into that camp.
I don't have any issues with my heat pump (rental property) or mini-split (primary home), but am always looking for ways to improve efficiency. Both units are LG inverter types with no backup resistance heating and blow nice warm air. The heating portion of the electric bill was about $120 last month for our house, which is comparable to what most people in an equivalent house would spend for natural gas in our area, but we have no natural gas line available. I really like my units and try to convince people that new heat pumps designs don't suck--- generally the hvac technicians/salespeople do.
I question the refrigerator analogy. If it were more portable and wasn't so heavy, it would definitely be cheaper to run if I could stick it outside at night. Right now in winter, it wouldn't need to be running at all with freezing temps. Inside the house, it typically only runs 25% of the time, unless you put a bunch of warm stuff inside or forget the door open. Somehow I'd like to logically circle this back to the idea for the heat pump.
A properly sized heat pump should run 100% of the time in the worst temperature conditions, which happen in only 1% of the lowest temps. So on average for daytime temps, it will be oversized by probably 2-4x. For example, in my area, the 1% lowest design temperature is 15F, and we might see a few hours of extreme lows around 5F through the winter, but more typically 20-30F overnight, and 35-45F during the afternoon. If it would spend more time running in the day, rather than night, my guess is that it should use less energy overall. Since it would be greatly oversized for typical daytime temps, I think it should easily get the house back up 5 degrees or so running for a few hours before everyone gets back home from work/school.
My guess was that running at 100% capacity for a few hours in the most ideal outdoor temperature should have the unit running at optimal efficiency, but that could be wrong. I'd like to hear a better argument as to why, and more importantly, where the sweet spot is. It might be something like running at 50% capacity, in which case, can a smart thermostat be programmed to run the heat pump at its optimal output and can it calculate what that is?