I took your idea of gathering the deltas of POTUS tweets, and just stuck to one month because it was tedious. Nothing jumps out at me, seems pretty statistically normal
To clarify, X axis represents number of minutes between a new tweet, Y axis represents number of tweets with that time difference.
Multiple tweets were counted multiple times, for instance three tweets at times 23:59, 23:56, 23:54 would provide a tick for 3 minutes, 2 minutes, and 5 minutes