General Relationship between Temperature and Barometric Pressure
The higher the temperature of the air, the faster it rises and loses density, decreasing the barometric pressure. The cooler it is, the denser the air becomes. This causes the barometric pressure to rise. Hotter temperatures generally mean lower pressure. This direct relationship doesn’t hold true for every instance of temperature vs. barometer.
On a global scale, the sun heats up the tropics, which causes air to rise. This rising air is the reason for the prevalence of cumulus clouds and thunderstorms in the tropics. Air flows in to replace it, causing the trade winds, while the air that rises blows north and south and sinks at about 30 degrees north and south latitude, creating high pressure zones in those areas. From there, some of the air continues on to the poles. This is the basis for all global weather patterns.
This process is further complicated by upper air winds converging and diverging. Converging winds force air down, creating areas of high pressure at the surface. Conversely, when upper atmospheric winds spread out, air from the ground rises and creates low-pressure areas.
On the smaller scale, an example of this relationship can be seen at the coast. Land is heated up faster than water by the sun, thereby lowering the pressure more over the land than at sea. This causes the cooler air, which tends to have higher pressure, to move in to replace the rising air over land. Sea breezes are the result of this process.
Local Temperatures Don’t Necessarily Fall/Rise Because the Barometer Is Low/High
Now, for any given location, the direct relationship between temperature and barometric pressure is more complicated. Were the temperature to drop from 70 degrees to 50 degrees in New York, for example, the air pressure would probably go higher. This is not necessary because the temperature changed, but because an atmospheric region of high pressure followed a path over the area. The same goes for low pressure areas and storms. Air pressure drops because the storm is passing overhead and not because the temperature has changed.
Although the core of an extratropical cyclone is colder than the surrounding area, in a hurricane, the low pressure center is warmer. This is because of the nature of how a hurricane is formed from a single air mass and the latent heat created by the thunderstorms. Extratropical storms form at the confluence of warm and cold air. The temperature difference intensifies the storm. This just provides an example of how, despite the large scale effects of temperature on barometric pressure, local processes can complicate that perception.