We have just purchased a second home on the coast, and in the winter plan to be there about 3-4 days out of each 2-3 weeks. We do have family living nearby who can turn system (heat pump) off/on and adjust the thermostat for very cold weather, but have been advised by a few to keep temperature set at 55 all during winter "to avoid high bills... that it costs more to bring it up to comfort level than to leave it set at 55 degrees all the time" It makes no sense to me to run a system when no one is there, but I want to be efficient. If this is the wise thing to do, would we also do the same with air conditioning, i.e. leaving it at a certain temp ,running while we are not there? Please advise!

For the way you plan to use this place that would be the way to do it. Of course when you are there bring it to comfortable temps