Cheap outdoor temperature running on power supply, not on battery?

Why?

Making a good outdoor power supply is “hard”, so they aren’t cheap. Connecting such a power supply to a sensor device in a way that will prevent moisture ingress is also “hard”, so also not cheap.

In my experience, cheap battery-powered temperature/environment sensors like the Aqara devices work very well outside, as long as you shield them from the elements a bit and use decent batteries (so no 10-for-€1 Aliexpress deals).