By the time of the events of "The Mule," history seems to have already forgotten him, even on Trantor itself, which became an underdeveloped agrarian backwater shortly after the Great Sack. Dagobert IX claims to have heard that Gilmer is dead by this time, though it should be noted that the rump-Emperor's grip on reality is badly eroded at that point.
Gilmer in "Trantor Falls"Edit
Gilmer, who proclaimed himself Gilmer I, Emperor of the Galaxy, was an interstellar warlord. He is the only warlord ever to attack and sack Trantor successfully: Before his time, the imperial military was powerful enough to defeat all foreign and domestic enemies well before they reached the capital, and after his time Trantor was too impoverished to present an enticing target. The latter was true because Gilmer was far more interested in looting the capital of its riches than in governing either it or the empire over which he claimed sovereignty.
Following his sack of Trantor, he was annoyed that the students and faculty of the Galactic University continued to resist his forces. He considered destroying the university from orbit but decided that an emperor should make a show of concern for cultural heritage and instead offered to negotiate a ceasefire with Dean Yokim Sarns. During their meeting, Gilmer surprised Sarns and the Speakers of the Second Foundation by his personal courage, entering their territory without any bodyguards. Much more predictably, he showed himself to be ignorant and brutish, with no belief in the value of learning, and was honestly perplexed that anyone could prefer the scholarly life to the satisfaction of combat--until Sarns explained that education allowed a society to make war more effectively.
Gilmer joined the Speakers (of whose identity as such he was, of course, ignorant) on a tour of the University and the Imperial Library. He was so intimidated by the accumulated learning on display in the Library that he agreed to an extremely generous ceasefire with the University without being mentalically conditioned, another pleasant surprise for the Speakers. A third pleasant surprise for them was that he had so thoroughly looted the riches of Trantor that it was no longer an attractive target for other warlords, meaning the Second Foundation would not have to endure another sack, though Hari Seldon had calculated a 70% chance that their planet would be subjected to at least two.
The Asimov estate considers those stories from Foundation's Friends, which can be integrated into the Foundation series, to be part of that series's timeline. Turtledove's development of the Gilmer character is arguably definitive.