1999 Ford Contour water light

The water light on my dashboard comes on when I start the car. Sometimes it goes off right away and sometimes it stays on. I've checked the water level and it's fine. Doesn't seem to be leaking any where, and the heat is working fine. On colder mornings, the l;ight will stay on longer. Also if the car is pointed down hill, the light doesn't come on. Could there be a water pump problem?
When the light stays on, I have to drive about a mile and the light will go out.