The three volume collected works of Gödel were financed and edited by the mathematical logic community, which was naturally not that interested in his hobby with relativistic cosmology. Although the Gödel chronology does list his intentions to study physics those first 2 years after his arrival at the University of Vienna in 1924 before switching to mathematics, there is no mention in the introductory biography that he learned about rotation in general relativity from Hans Thirring. [The editor-in-chief Solomon Feferman should not be slighted for not knowing this since the editorial note and short bio introducing the reprints of Gödel's 2 relativity papers in General Relativity and Gravitation also did not report this "fact", found by an internet search.] Thirring and Josef Lense had studied the field of a rotating body around 1920, whose classic work on this topic was recently reprinted in English translation . In an unpublished lecture at the Institute for Advanced Study in 1949 made available in his collected works [vol. 3], Gödel remarks that ``a note by George Gamov in Nature  suggested that the whole universe might be in a state of uniform rotation and that this rotation might explain the observed rotation of the galactic systems," which must have inspired his original work in relativistic cosmology presented a few years later.
The followup biography by the third volume collected works co-editor John Dawson Logical Dilemmas: The Life and Work of Kurt Gödel does better, referring to Thirring as Gödel's former physics teacher and notes that Thirring was a member of his postdoctoral thesis (Habilitationsshrift) committee. It also describes the period 1946-1949 in which Gödel did his cosmology work in some detail in chapter 9 (including the mention of his famous notebooks on the distribution of galactic rotation) and does give some background about how this interest arose apart from the catalyzing request in 1946 for a contribution to a volume to honor his friend Einstein. The physicist Hans Thirring is also mentioned in chapter 2 as having belonged to a committee with Gödel's dissertation director Hans Hahn to investigate "mediums" (parapsychology), together with that another important mathematician Moritz Schlick in this University of Vienna group who occupied the chair in Philosophy of Inductive Sciences, once held briefly by Ernst Mach, a physicist (Mach number and Mach's principle), mathematician, psychologist and philosopher whose principle was shaken up by Gödel's later work.
John Wheeler in his autobiography Geons, Black Holes & Quantum Foam (1998) tells an anecdote about meeting Gödel in 1971 or 1972 with his Gravitation coauthors Charles Misner and Kip Thorne during which Gödel was very eager to talk about the distribution of rotation among the galaxies.
Engelbert Schücking [physicist, relativist at NYU, 2000] tells of the following encounter with Gödel in 1961 regarding his use of the work of the Italian mathematician Luigi Bianchi used by Gödel in both of his papers on relativistic cosmology:
"H.P. Robertson at Princeton knew about the Bianchi types. He mentions Fubini in a footnote to the cosmology paper in Reviews of Modern Physics, vol. 5, 1933, which led me to Bianchi. And that was how I learned about them before I saw Taub's paper. Did Taub learn from Gödel? Gödel had done his work in 1949 and published some results without proof. I reconstructed some of the proofs and visited Gödel in 1961 in Princeton to find out whether he was going to publish his results. Especially I wanted to know whether he had a clever way to avoid doing all the different types separately. Gödel showed me a full filing cabinet with calculations. He had done all the types with matter separately. When asked when he would publish he told me: "not in the next ten years." These calculations have never been published. Could Gödel have suggested to Taub to do the vacuum cases? Unfortunately I never got round to ask Abe about the history of his great paper..."
[My one and only lunch in the Princeton University Prospect Faculty Club before 1999 had been as an undergraduate in 1973 with Schöcking and then assistant professor Remo Ruffini when the former had come to give a black hole talk at the physics department. Fubini had been a student of Bianchi, eulogizing him after Bianchi's death in 1928, and had passed through the Princeton Mathematics Community story in the late 1930s.]
I met Gödel at the Institute in 1973 when I was an undergraduate at Princeton University to discuss rotating cosmologies for my senior thesis, and he informed me of the latest current work being done in the field by Michael P. Ryan, a former student of Charles Misner, that as a immature college student, I was still unaware of.
Robert T Jantzen, Princeton '74 (physics)
The Princeton Mathematics Community in the 1930s