So I threw together some (rather generic) desert Gym designs. None of them are really intended for certain Types, but some would probably fit better than others. I also haven't created leader skins or something (yet?).
1: Hidden Oasis Gym
This one is a hole in the desert floor, filled with a spring and two little water pools. Some vegetation managed to grow into a quite green oasis here, shielding itself from sandstorms with a roof of acacia leaves. Several sandstone spikes form a loose open dome shape.
Down in the oasis are three rooms where trainers and leader can be placed (alone with a few along the path).
This one could be Grass, Bug, Poison, Psychic, Normal or Fairy.
I just chose the Heracross statues because they fit the desert theme (ancient Egypt's obsession with scarabs) but can be changed to Pokémon fitting the Gyms type.
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2: Roman-Egyptian Temple Gym
Loosely based on Roman architecture and Egyptian buildings that were inspired by that. Trainers can be placed along the colonnades or in the central courtyard. The leader waits in the central building.
This one can be any type. The golden Arbok statue was chosen for style (the cobra was associated with the Pharaoh), but again can be exchanged for a more fitting Pokémon.
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3: Rock Spire Gym
A giant rock protrudes from the desert here, mysteriously sprouting water from its side. The leader and their friends built platforms onto the sides of the rock and use it for their Gym.
This one might be good for a Flying Gym, but Psychic, Fighting, Rock, Normal or Dragon would work as well.
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4: Underground Gym
Rather plain at first sight nearly all of this Gym is underground. Only the entrance gives it away.
Inside is a small maze where next to a few statues of Pokémon (which can be adapted as always) there is a closed piston door. It can be opened with a lever that is found in another part of the maze. Behind it you will enter the crypt the leader awaits you in.
I'd suggest Dark or Ghost type, but other others like Rock, Ground, Normal, Steel or Poison could work as well.
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Let me hear what you think about them. :D