In most applications it saves energy to turn it down. *However if you
have a heat pump, and to get the house warmed up again it goes to
emergency heat, then it can cost more. *If you can turn off the
emergency heat and wait a long time for the heat pump to catch up, then
you will save energy. *The worst case is electric emergency heat, gas
emergency heat may or may not save money depending on the price and
efficiency of the furnace. *

The above is the CORRECT answer... congratulations!!

Mark