We have a gas backup heat on our heat pump.Our outdoor unit probably needs replaced but I was going to put it off until the first of the year if possible.Is it safe to run my heat pump in emergency heat for an extended period.

That should not be a problem. Most peoples' definition of emergency is "resistance heating" which is EXPENSIVE to operate. You appear to have a dual fuel system. Stage 1 is likely HP and Stage 2 heating is gas and emergency.

Sure,, its safe,, people that don`t have heatpumps use a gas or elect,or oil furnace,, the same as what you have,, just expect to pay more for utilitys