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The Empire are often viewed as the bad guys in Star Wars, and not without reason. They use the Death Star to completely destroy the planet of Alderaan in the blink of an eye, and have been depicted as subjugating millions of people in the galaxy amongst all of Star Wars media.
However, the Empire had to maintain its control throughout the galaxy, and this couldn't be done if they were treating every planet badly. Because of this, quite a few planets actually prospered under the Empire, which is something only die-hard fans know about. These were typically planets located within the Inner Rim, that could offer the Empire enough in return. With this kind of prosperity, it's easy to see why the Empire did have its supporters.
10 Lianna
Lianna, an Outer Rim planet, was surprisingly important to the Empire and actually prospered under its rule. The planet's ruling house was a staunch supporter of the Empire, retaining full support until Emperor Palpatine's death when they switched sides to the Republic. Despite this clear favoring of the Empire, the New Republic chose to lead by example and didn't punish them for this, even though an Imperial remnant presence was situated there.
Lianna provided the Empire with many of its TIE fighters, and the Royal house vocally supported Emperor Palpatine. Because of this, the planet became fairly rich from supplying the Empire with their fighter ships and supporting its doings.
9 Corulag
Corulag had the advantage of being a Core World planet, meaning it had already managed to ascend to a favorable position within the Galactic Senate during the days of the Republic. In the Empire, the planet managed to retain this position. An Imperial Academy was installed on Corulag, which provided the Empire with future generations of Storm Troopers, ISB Agents, and more. The Imperial Academies were one of Jedi Fallen Order's best things, and many fans feel they need expansion in future media.
With the planet facilitating this training and a place for prospective Imperials to learn, this inevitably elevated the planet to having more overall importance to them. During the days of the Empire, Corulag would find itself quite protected against neighboring planets.
8 Carida
Carida is a mountainous planet located in the Colonies section of the galaxy. During the Empire's reign, Carida would become quite an important world, as it was the home of an Imperial Training Academy as well as a Naval Docking Facility.
It was a place for the future generations of Storm Troopers to train, as well as a place for the Imperials' ships to refuel before heading back out on patrols. Thus, Carida was well protected due to its helpfulness to the Empire.
7 Corellia
While Corellia was depicted as a smoke-stained industrial wasteland in Solo: A Star Wars Story, the planet's living conditions actually improved under the Empire. A ship-building world, the Empire gained many Star Destroyers and other vehicle-class starships from the Corellian people.
Corellia's usefulness to the Empire gave it a certain level of protection not afforded to other planets. A lot of people on the planet within the ship-building industry also had valuable connections to high ranking Imperials.
6 Berchest
Berchest was located in the galaxy's Inner Rim, giving it a favorable position seeing as the Empire tended to have the most control and presence in planets closer to the Core. The planet was the site where many Imperial yachts refueled and parked, meaning that lots of high ranking Imperials and rich citizens of the galaxy had business on the planet.
This association with the elite meant that Berchest was relatively well protected, and employment opportunities were high. While the planet may not have been the most important to the Imperials, it still had some use and was treated accordingly.
5 Naboo
While there were downsides to Naboo during the Empire, such as demilitarization of their native forces and Imperial military takeover, the planet actually prospered during the age of the Empire due to both its location and its status as Palpatine's home world. Naboo featured in the first film of the prequel trilogy, The Phantom Menace, and has been expanded upon ever since.
Naboo's economy and environment were considered far healthier than most of the other worlds in the galaxy. It also stationed a military garrison for the Empire, giving it a useful status to the Imperials. This meant that the planet was well protected, and well stocked in terms of resources and food. It presented a lot of more opportunities for the average person, rather than more neglected planets such as those found in the Outer Rim.
4 Coruscant
Coruscant already had status, having been the capital of the Republic for many centuries. This capital status did not change when the Empire took over, and it became the site of many Imperial political buildings. This meant that the planet still had great influence over the many other worlds in the galaxy. A lot of the prequel films' best characters lived on Coruscant, making it a very important planet to the series.
Whether Coruscant prospered or worsened during the Empire depends on the point of view. The disparity in classes on Coruscant was still largely present, but conditions for the average person did improve as the Empire favored planets closer to the Core and thus gave them more resources.
3 Thyferra
Thyferra was not the site of any military buildings or shipbuilding areas like the other planets that prospered under the Empire, however, it did have great usefulness to the military organization. Thyferra was the source of Bacta, a super healing substance that could be placed into Bacta tanks to facilitate fast healing. Fans of The Book of Boba Fett will recognize this substance as one that Boba uses to heal.
Bacta was incredibly useful to the Empire in order to keep their soldiers healthy and able to recover from battles, and since the Empire got into combat missions all across the galaxy, the need for it only increased. This meant that Thyferra was treated much better during the Empire's rule, and living conditions were higher for the average person than they were on other planets.
2 Eriadu
The home world of Grand Moff Tarkin (one of the best original trilogy characters), Eriadu prospered under the Empire's rule. Living conditions for the average person improved, despite being fairly high to begin with. An Outer Rim world, Eriadu had a mixture of different terrains, which had gifted it many resources that helped it prosper long before the Empire was established.
The mining operations provided the planet with much of its trade and allowed it to amass enough capital to earn a stronger political seat in the Galactic Senate. Tarkin's rank and influence within the Empire also afforded Eriadu with special treatment.
1 Kuat
Kuat was a Core World that had already enjoyed great influence during the days of the Republic. In the Age of the Empire, the planet's importance only increased. It became a vital shipbuilding world for the Empire, and because of that, it prospered under the Empire's reign. Fans of The Mandalorian who enjoyed Kuill's memes will know that he comes from Kuat.
Home of the Uggnaughts, Kuat provided the Empire with many of their warships. This meant that the Empire had to keep its citizens healthy and happy, in order to ensure that productivity remained high.