4000 DA is 4000 DA whether the temperature is 0F or 120F. ... That would indicate that that at higher temps in any given DA, the air density increase.
Not exactly, the performance of the bullet was usually calculated in a given set of conditions for the BC, with a minimal impact of the DA in the equation.
You are making corrections with the Density Altitude for the air conditions the bullet will be fired into, not just the physical altitude and ambient temps for the location of the shot.
The DA is calculated for the factors of barometric pressure, the altitude, current temperature, and the humidity - all which influence air density, and cause the bullet to act as if the altitude were different.
The temperature frequently has a significant impact on the DA, you are seeing the increase in corrections because the given set of conditions you are looking at will cause the bullet to act as if it was being fired at a lower elevation/temperature.
This is much harder to explain in shooting terms, than pilotage...as a long time helo guy, we spent hours calculating the effects of the DA on our aircraft performance, by time of day, location at start/during/end of mission, and the load plan for the flight.
But it was much the same, we had plan where the aircraft "thought" it was flying, more than just the current altitude/temperature.