Obviously with a given mass and volume of air, pressure is directly proportional to temperature. However, I would expect the total mass of air within a column of atmosphere to vary over time, due to atmospheric waves (and oceanic waves, over the ocean).
How highly correlated is surface air pressure with surface temperature, at a given site? How much does the variation of atmospheric mass affect that relationship?
You have opened a very large can of worms. You will need a larger can to get them all together again.
The problem is time lag.
Let's do some thought experiments:
It's dawn in the desert. Right now the bottom of the atmosphere is cool.
Sun heats the dirt, and the dirt heats the air. Air expands some. But the air above it is lazy about getting out of the way. So the air pressure goes up some. Air finally shoves that air above up, and then whether the pressure goes up or down depends on how fast it can get out of the way. Small area -- parking lot, the air spills in all directions. adjusts quickly. Gobi desert air starts to push out sideways, Coriolis rears up, and you have a temporary high pressure. At night the reverse happens. Ground cools, air shrinks, but the air around it doesn't fill in right away.
Lets look at longer term features. Consider the Alaska low. The north pacific is warm, at least by ocean standards, and so the air above it is warm. Warmer air is less dense, and since the low stays more or less put for months at a time, the entire air column is warmish. Warm air is less dense than the cooler air around it. It has a lower effective surface pressure than cooler air around it. The cooler air from around it tries to rush in, but... Coriolis again. Friction restricts how much of that air can enter the low area. Low pressure warm air is pushed out by moving upward as a series of cloud and rainy weather.
We think of the atmosphere in our heads as this thick blanket covering the earth. But most of the interesting stuff is in the bottom 3/4 of the atmosphere (by mass) which is about 36,000 feet. Call it 5 miles. It's a thin apple skin.
One of the messy bits is the scale problem: There is very little change in energy moving a parcel horizontally. But moving it vertically gets messy. Pressure changes. Temperature changes. Water condenses/evaporates adding/removing sensible heat.
One of the eye openers for me was realizing that in a lot of physics, you can analyze and quickly conclude that certain forces have small effects and can be ignored. In atmospheric dynamics I find myself unable to figure out what can be ignored, or left with a first level approximation. Everything affects everything else.
Anyway to answer your question: Warming temperatures will initially increase pressure, but later decrease pressure. Cooling temperatures the opposite.
Looking at Edmonton, AB's forecast for the next 10 days, we are getting daily temp fluctuations of about 11 C, and daily pressure variations of about 4 hPa.
From the gas laws this is an 11 K compared to about 300 K so the pressure should change by about 11/300 of it's absolute pressure. This would be about 37 hPa. Why doesn't it?
Because it's not the whole air column that's changing temp. Typically only about the bottom 3000 feet changes on a daily basis -- daytime mixing depth, with some mixing higher up. 3000 feet has about 1/10 of the atmosphere in it so the variation is only about a tenth. Ok, that gives 3.7 hpa. Closer to 4, but still off. Remember you ignore factors at your peril in atmospheric.
