AIUI, air temperature relates almost exclusively to the qualities of the air mass that is being measured.
An air mass is a parcel of air sharing relatively uniform properties. The temperature of the air mass is mostly affected by where it came from, how far it has travelled and what it has travelled over - arctic air masses will start off cold but this can be moderated by passing south over warm seas for example.
In this instance, very small pressure differences (in the scheme of things) doesn't have a massive difference on the temperature of the air. The air isn't in a solid container so it can never be exposed to the compressing forces necessary to alter the temperature by any meaningful amount. Meteorologically speaking areas of high pressure or low pressure really are referring to airmasses which are either sinking or rising.
But I am by no means an expert so it's all probably rubbish.