The show deliberately leaves the lore and theology of the Tuttle Cult open to interpretation.
I like imagining that True Detective’s first season literally takes place within the canon of the Cthulhu Mythos, and that someone involved in founding the cult had a brief, partial encounter with Hastur (a.k.a the Yellow King) and the dream-realm of Carcosa, with the writers keeping it tasteful by avoiding open supernatural events and having the detectives deal with the material, non-supernatural mess the cult’s worship left behind, and the cult itself being one of possibly many ramifications and of how the existence of the Great Old Ones and Outer Gods reverberates into human society.
My idea is that the cult’s founders encountered something, possibly through psychedelic drugs or another altered state, that they barely understood, and then they built their beliefs around it, and later generations inherited a mixture of that experience, distorted explanations, rituals, and abuse. They encountered one small part of a much larger reality and made it the basis of everything they believed. In Call of Cthulhu, people receive impressions through dreams without understanding their source. Hypnos explores altered states and access to things beyond ordinary perception. A person can experience enough to become obsessed while still having almost no understanding of what happened. Of course, Chambers and Bierce created Hastur and Carcosa, and Lovecraft only name-dropped them in another book, so the link to the Mythos mostly comes through later writers like Derleth, but the point stands that Hastur exists as a lovecraftian entity within the mythos, which is what my theory is based on.
There's also a lot of philosophical and thematic ressonance between the show and cosmic horror, which is something a lot of people point out. Rust’s pessimism and some of the things LeDoux and Childress say fit comfortably into that existencial horror that the works of Lovecraft and other cosmic horror writers lean upon. The existence of the Old Ones and Outer Gods are powerful because it summons within the reader this melancholic thalassophobic feeling of powerlessness and nihilism in the face of uncaring entities within a dark amoral reality, not unlike the vibe I get when I watch the show. The antagonists in the show commit evils so intense and so hard to comprehend that their methods and values come across as almost alien, which echoes the way that the existence of the Old Ones and Outer Gods often are so unknowingly complex and evil that it breaks the human minds of those who glimpse it.
Finally, and this is kind of a minor point, but usually when a story is based on another medium, its world usually leaves that medium out of the world that story takes place. The MCU has no Marvel Comics company, no Stan Lee, no comic books about its heroes, etc. The characters in The Walking Dead never say the word “zombie” because zombie movies do not exist in their world, etc. Rust and Marty are capable, well-read investigators, and they never come across a copy of The King in Yellow or any Lovecraft story. If those books existed in their world, the detectives would at least have found a mention of them. Instead the show treats these ideas as completely unknown. Rust becomes obsessed with understanding them, and the detectives go to absurd lengths to piece them together from scratch. That only makes sense if the books do not exist in the show’s reality. I personally like to fill in that gap with the possibility that the events are happening in the same reality as those stories.
This is just my own headcanon, based on my personal taste in horror and what makes the story even more interesting to me, as a fan. I’m sharing it for the fun of discussing that possibility but obviously I'm making no claim about the writers’ actual intentions or the objective truth of the show.