Does the pimple necessarily have to be warm to be considered infected?

Not always: Most common pimples are caused by bacteria, and are therefore, by definition, infected. Infection USUALLY results in inflammation causing 1) redness, 2) swelling, 3) tenderness, and 4) warmth. However, unless the pimple is big enough, or your measuring is sensitive enough, it might be hard to tell it is actually warmer than the surrounding tissue.