I think it would depend on the particular gazebo, the horse(s) involved and the conditions it was used in. I've seen them used as shade at competitions for horses tied next to trailers/wagons but I wouldn't leave one in a field with anything who like scratching or playing. Some of the pop-ups are pretty robust but they're not cheap either - generally the cheap ones snap if you sneeze underneath them!
Just looking at the one I have outside on my decking atm, I would say, No. Ours is a fairly robust Ikea one and we put it up in April and leave it up til September, it sits across our decking and forms a nice little outdoor room for the summer. However, we do screw the poles down to the decking boards and use ties as well, so held down a bit like a tent. I would say that in a field, unless butted right up against a very high hedge on 2 sides it would take off at some point, or rip apart. The possibility for the ponies getting hooked up in the framework or spooked by flapping bits of material is too great imo. They are also quite bendy and a good butt rub or neck scratch would probably just twist the frame beyond repair. I did debate having one of those roll out ones fitted to my little lorry but again, I dont think they would be very robust and would worry about them if the wind got underneath.
Lol er nooooo.Solution- 4x spiked met posts,4 x8ft lengths of 4x4.Drive met posts into the ground 12 foot apart to make a square then place 1 length of 4x4 in each.Lenghts of 2x3 to form another square around the top plus two across the middle to act as rafters.Cover with roofing material of your choice.We used galvanised sheet.Sturdy permanant shelter for around £200 or much less if you can use salvaged timber.
Or four posts as above with a shade sail attached and then in the less than sunny weather the sail can be packed away and haynets can be hung on the posts for winter