When you fill up a cabin with warm air the pressure of the cabin will rise. An outflow valve controls the desired pressure. But with pressure rising doesn't the temperature rise because you're compressing air into a space? Is this why the outflow valve is there, to regulate the pressure and temperature? Does the outflow value open and close depending on the pressure during phases like climb, cruise, and descent. Or does the outflow valve shut completely once the cabin altitude reaches 8,000ft? But then wouldn't the air get too hot because it keeps getting shoved into the cabin? Does it use the isothermal rule, if so then what is the work done?
Pressure does not rise because keeping the cabin pressurized begins with releasing air to the outside. The flight starts low, with low cabin altitude, as the plane climbs the cabin altitude rises (lower pressure than it started).
As for the air itself. When it is compressed by the ACM, the temperature does rise, that's why it is cooled before being introduced into the cabin.
Work is done on the air, then the air does work by passing through a heat exchanger, and then a turbine, which runs the ACM.
There is one plane I know of that maintains sea level pressure up to FL 410. In that case air will still be jettisoned overboard to bring in fresh air, with the same ACM cycle of temperature regulation.
When you fill up a cabin with warm air the pressure of the cabin will rise.
You're not filling up a cabin with hot air, you're allowing sufficient air (of any temperature) into the cabin to match the air flowing out. What is important is the flow of air (~10 pounds per person per minute).
An outflow valve controls the desired pressure. But with pressure rising doesn't the temperature rise because you're compressing air into a space?
No, the outflow valve makes sure that the pressure DOESN'T rise, it stays roughly the same (allowing for climb/descent).
The classic paradigm for aircraft pressurisation is a (toy) balloon with a pin-prick leak - as long as you keep blowing it up as fast as the air is flowing oyut the leak, it will stay inflated. The outflow valves use flow as a means of controlling the pressure (differential), which then ensures the requirement for sufficient flow of breathing air per passenger.