underground. the temp stays constant at a certain point . after that it must increase at what depth does it?

Question

increase and what depth does it stay at the level used for homes cooling and heating.

Answer 1

That depends on the geothermal gradient (called "Q") at your location.

In general, the thinner the earth's crust, the higher the gradient, and it will get warm fairly close to the surface. Examples are the Eastern Rift in Tanzania and the Great Basin in Nevada; both of these areas have active hot springs. In some areas the the bedrock can get measurably warmer only a few meters below surface.

Areas of older stable rocks (called cratons) such as the Baltic Shield in Finland-Sweden and the Canadian shield in Minnesota and Ontario have very low geothermal gradient, where you'd have to go 100's of meters below the surface before the temperature increases.

Most ground-source geothermal systems are within 2 meters (6 feet) of the surface due to the expense of putting it any deeper. Another option, called water-source, uses deep wells for the heating and cooling effects. In general, the deeper you go the more stable the temperature will be, as there will be less variations due to hourly, daily, and seasonal influence from the atmosphere.

Answer 2

Temperatures increase 'typically' ~25 degrees celsius per 1 km increase in depth.