Yes the temperature, winds, and pressures are all connected. As the winds weaken and reverse direction during a major sudden stratospheric warming, there is a component of the air that moves poleward and descends rapidly over the Arctic and pressure increases. As the air descends it warms: this is one of the reasons why the temperatures can increase so impressively during a major warming event. This can be seen through the lower stratosphere but its connection to the troposphere is not as well understood. Typically a sudden stratospheric warming will increase the likelihood of the negative phase of the Arctic Oscillation which is associated with an increase in pressure over the Arctic and a shift of the tropospheric jet stream.