After reading 'The King in Yellow', I think the series more closely addresses the views of Robert W. Chambers (and Ambrose Bierce) than H.P. Lovecraft. I myself love both Lovecraft and Chambers equally, but Lovecraft was an atheist who considered the cosmos at large to be mechanical and material, and denied the existence of "higher" beings. Conversely, Chambers thought there were means by which people might know absolute truth; he wrote about the contrast between nature and reality (which he believed to be God) and the possibility of seeing beyond the illusion of this "reality" via art to encounter a truer God or devil, represented by the character of the King in Yellow, the danger being that art may be but illusion created by the madness of man; these represented the two competing perspectives within his own soul: his love of art and his love of nature.
In 'The King in Yellow', we learn about the mysterious King in Yellow, who lives in the surreal world of Carcosa, which has black stars and seems to operate outside of the laws of nature; it might even be said that Carcosa seems to operate according to the laws of art as a remedy of sorts to the "tyranny" of natural law, but the reader is free to hold that the King may be a greater tyrant. His name-sake play seems to share "truths" with people, but these truths generally drive them mad or cause strangeness to affect their lives.
Conversely, 'Hastur' is a, at least seemingly, benevolent if mysterious god of shepards who was invented by Ambrose Bierce and referenced in Chambers's book; I strongly believe that understanding Hastur as described by Bierce is central to understanding Chambers's invocation of the character in his book, an idea which, both by name and by theme, serves as a counterpoint to the King; while the King is the central theme of the early stories, the later stories focus upon the weirdness and "magic" of real life and nature, and this is no coincidence While he is mentioned in connection to the King's play, Hastur seems associated with certain real stars such as Aldebaran and the Hyades, and is hinted that he might even be a star himself. Hastur even makes an appearance in Chambers's story 'The Demoiselle D'Ys'; in the story, while seemingly connected with certain animals like falcons, he doesn't make an attempt to stand out and takes the appearance of a well-dressed, humble servant (a stark contrast to the King in Yellow who is described as a king who wears tatters). Similar emphasis upon life and nature occupy the remainder of the book and, in contrast to the first part which emphasizes the otherworldly glimpses ascertained through art, puts a lot of emphasis upon discovery through others, experience, and nature, taking the supernatural to be found in these.
In a similar way, I think the first season of 'True Detective' continues this dichotomy between art and nature set up by Chambers's 'The King in Yellow'; it's probably no accident that the serial killer slays people and animals to create perverse art, trying to gain entry into Carcosa. Rust Cohle, with his views, seems closer to the way of The King in Yellow. In contrast, Marty puts his faith in his experience of the world (nature) rather than ideas, for better or worse trusting in the mysterious higher power he sees as guiding everything through nature. Therefore, these two characters convey to the audience the contrast between the King in Yellow and Hastur: Carcosa and "living" space-time.