It's an inefficient machine and much of the energy is dissipated as heat rather than performing its intended function of inflating the tube, perhaps, Sandy. But what happened to the vintage Roller?
These are straitened times and just for now the motor is up on blocks. Besides, cycling down to the newsagents now and then allows me to think I'm doing my bit for the environment.
The equation PV=nRT, describes the ideal gas law.
Simply stated, when any gas' pressure is increased, it's temperature is increased. Conversely, when the same volume of gas' pressure is decreased it's temperature is decreased... hence the cold air when deflating your said tire... according to Boyle, Charles and Avogadro
Could it also not have something to do with the air being sucked in at a great rate at the top of the pump when the handle is drawn up, that warms the pumps?
//IN layman's terms it's the compression that generates the heat?//
The heat is already there. The energy present (in the volume of air) in being confined to a smaller space results in an increased pressure and a corresponding rise in temperature.