The amount of gas produced is not linear with temp. The amount of gas produced is dependant upon propellant charge.
The pressure of that gas is linear with temperature.
Lots of external factors effect pressure/temp during ignition of a powder charge. The most important factor being barrel steel temperature. Compare the ignition cycle of the same rifle at two extrem temperatures. If a rifle is minus 100 degrees Celcius when it is fired, the barrel will suck many calories from the chemical reaction of the propellant, muzzle velocity will be reduced because heat was robbed from the reaction and lowered chamber pressure.
Compared to the same rifle fired at 200 degrees celcius. The barrel steel is three hundred degrees warmer, and while it is still much cooler than flame temperature of the powder charge, it will absorb much less heat from the reaction than the colder barrèl. Possibly allowing temperature and pressure to elevate far enough to become dangerous.
All of us have seen the variatons in muzzle velocity produced by shooting in varied temperatures. While we obviously do not shoot in the extreme conditions I portrayed, I have test fired rifles from minus 15 F to 110 degrees F. The differences in muzzle velocity are easily recorded and are solely dependant upon changes in chamber pressure.
If powder A produces the same velocity as powder B with powder A having a lower temperature, powder A also delivers a lower pressure. There are many examples of this., because the duration of powder B is longer.
Take for example the 30-06 with a 150 gr bullet. From Hodgdons #26, H322 at 46 gr will produce 2720 fps at 50700 cup. While H414 at 58 gr will produce 3043 fps at 48700 cup. H322 is the hotter of the two powders. It produced the highest temperature and highest chamber pressure. But H414 produces more gas because it uses more powder to start with. It produces more velocity at lower temperature because the pressure, although lower, is sustained over a greater period of time
