I should begin by stating that I have no special knowledge of the secret doctrines of Scientology, and I do not know what the group teaches its followers beyond the publicly available information that has been widely reported since its disclosure during legal proceedings in the 1980s. The New Yorker recently ran a major story about Scientology (Feb. 14, 2011), and reported what the court document and news accounts of the era had made public: that Hubbard claimed an ancient astronaut named Xenu came to earth 75 million years ago and buried a billion or more aliens beneath volcanoes and killed them with hydrogen bombs. Their souls (or thetans) are said to now infest human hosts, causing many problems.
There are some superficial similarities between Lovecraft’s and Hubbard’s visions of our alien past. Both wrote that extraterrestrials came to earth tens of millions of years ago, and both wrote that earth had been a part of a galactic system of inhabited worlds before a cataclysm caused the aliens to retreat. Both also wrote about buried evidence of alien civilizations (in Hubbard’s case, alien implant stations and Xenu’s prison), and both wrote about the ability of minds to travel millions or billions of years across time and millions or billions of miles across space for encounters with the aliens.
However, Hubbard’s vision is very different in detail and in tone. Lovecraft imagined a grand cosmos of aliens who were utterly inhuman and incomprehensible, who treat humans as elephants might treat earthworms. By contrast, Hubbard’s aliens are essentially human in all but name, possessed of human vices and motivations. Lovecraft’s cosmos is also much less dependent than Hubbard’s on the tropes of space opera and Golden Age science fiction (presuming, of course, you take Hubbard’s cosmology as a literary text rather than revelation).
It is a fact that Hubbard was a science fiction writer active in the same years that Lovecraft's stories were first published (the late 1930s—some Lovecraft tales were published posthumously) and writing for the same types of pulp magazines in which Lovecraft's stories appeared. However, the two authors’ outlets overlapped only at Astounding Stories (after 1938, Astounding Science-Fiction), the magazine that published At the Mountains of Madness in 1936. This story, however, includes the same type of cosmic sweep as Hubbard’s cosmology, though both approach the concept in very different ways. Hubbard developed Dianetics (the precursor of Scientology) for Astounding Science Fiction (1950), and science fiction luminaries such as L. Sprague de Camp and Astounding editor John W. Campbell were friends of Hubbard and also well-versed in Lovecraftian fiction.
It would go far beyond the evidence to suggest Hubbard borrowed his cosmology from Lovecraft, but the core concepts of ancient aliens, buried civilizations, and mental transfer across time are all ideas that Lovecraft wrote about in stories that Hubbard almost certainly would have read years before developing OT-III. Nevertheless, the reported revelations of OT-III are much more similar to Golden Age SF space opera influence than anything Lovecraft would have written.
I have previously established in The Cult of Alien Gods that Lovecraft was the primary force marrying Theosophy’s idea of planets inhabited by ascended masters and human souls waiting to be born (itself derived from medieval notions of planets as the seats of various ranks of angels) to science fiction's non-spiritual extraterrestrials in order to create the modern myth of ancient astronauts. In this limited sense, later works like Scientology’s OT-III (taken again as a literary text) can be thought of as influenced by the ancient astronaut myth Lovecraft developed in the 1920s and 1930s.