What does it mean if the temperature gauge reads cold all the time

My friend's 1996 Audi A6 has weird temperature problem and it beeps and displays an engine temp warning? The temperature gauge for the engine doesn't move past cold most of the time, but never goes hot; however, the panel beeps whenever the car accelerates or goes uphill, and the digital display shows the engine temp symbol. Also, when you open the hood, the coolant expansion tank is almost empty even though it was filled a month and a half ago with coolant.