why would it not be necessary to heat the air in the balloon as much if the air outside the balloon is colder?

Hot air is less dense than cold air and so hot air will be buoyant when surrounded by cold air. Hot air balloons can only fly when the air inside the balloon is much hotter than the air outside it. When the air outside is very cold, then the air inside the balloon doesn't have to be as hot to lift it up.

Hot-air balloons float because the air caught inside the balloon is heated up by a burner, making it less dense than the air outside. As the burner heats the air, it expands and some of the air escapes; that's what makes it less dense