I made a light frame and put hardware cloth on it. When the box is in use, it leans against the fence. When on the sandbox, it keeps cats out and the sun in. Houston Chaining a hungry Rottweiler to the sandbox is effective. If you could somehow breed those sand worms in your sandbox, that would take care of the cat problem, not to mention those pesky neighborhood kids.
Kids can scoop and bulldoze without the complimentary cat favors.
Gravel also doesn't track in the house as easily as sand. If all else fails, you could get one of those calf ropers from the rodeo and put him to good use. They don't have enough power in it to be a killer. There isn't a cat alive that will not remember a well-placed BB in the butt. If a cat has been using it, get all the sand out and replace it with clean sand.