The answer to this question depends on the gazebo’s structures. Permanent units made of metal or wood can be placed anywhere you want, on the ground, grass, cobbled stone, patio, or on concrete floors.
Foldable and ready-to-use units are usually installed on grass because they come with poles, similar to tents. The main downside is that they can only remain on the grass or sand as there is no way you can stick the poles into concrete or stone.
There is also the option of elevated units but this is mainly suitable for wooden constructions. These gazebos are usually raised on wooden planks so that they can be placed on any type of surface and be easy to transport from one place to another, without having to tear down the entire construction and rebuild it.
No matter your choice, make sure you don’t go for a cheap gazebo with plastic poles as these can easily break or bend.