Heat pump in question has no emergency heat installed. Location is
Phoenix, AZ.

Is it better to turn it down at night or not. User is on a time of
use plan from Salt River Project. Peak time in winter starts at 5am
but user gets up at 4am and can turn it on then.

Saw this in another group, is it true?

If one has an electric heat pump and is on a time of usage plan with
the power company there are two more considerations.

The power rate goes up at 5am so you are likely to be re-heating with
more expensive power. Also, a heat pump is less efficient as the
outside temperature drops. As it is usually coldest just at dawn,
kickiing the systm on then to re-heat is making it try to work when it
is at it's least efficient state