How deep do the footings have to be for a gazebo?
How deep is the frost line in your area? Deeper than that. You may also need to use post-footing-base flares to get adequate load capacity on the bottom of your footings, depending on the soil bearing capacity (what it can hold per square inch, in pounds) and the weight of the roof and any roof loads, such as snow.

This type of thing, possibly: 

 

Image is from bigfootsystems, but lots of places make something similar, and the old-fashioned way is to pour a pad in the bottom of the hole and then set the sonotube on top of it (no fancy plastic flare required.) In "good" (from a load-bearing point of view) soils you may not need anything, if the area of the sonotube (about 50 square inches for an 8" tube) is adequate for the soil and loads. Bumping up to 12" at the bottom more than doubles the area, thus halving the soil loading.