Aye, this one'll be complicated, Captain.
Lovecraft is the link between True Detective and Chambers's The King in Yellow. Lovecraft is the link between the figure of the police investigator and that of the artist-investigator. The first, uncertain tropes of weird fiction that precede Lovecraft's work reach new maturity in his stories. From there, the questions that both detective fiction and weird fiction raise are explored, without being answered, and passed on to successive generations of writers.
The inner ring:
The unreliable narrator. Good lord, where would Lovecraft be without the unreliable narrator? It's the centerpiece of his fiction. The details vary among stories, but Lovecraft's main character is often an investigator who (1) opens his story that he knows few if any will believe, and does so (2) when he is already on the edge of sanity as a result of the events he has witnessed, possibly in which he has participated, and whose story is thus (3) part scientific log, part investigative report, part confession.
The narrative palimpsest. In Lovecraft, the relatively simple notion of a palimpsest--a document that has had its original writings mostly or completely removed and another set of writings overlaid--becomes more complicated. There are multiple movers in multiple domains in Lovecraft, ranging from the unpredictable, lunatic ravings of gods and godlike figures, to the similarly lunatic worship conducted by their cultists, to the occasional victims of the cults who witness terrible events, to the investigators who for whatever reason catch wind of the cult activity and follow up, to the authorities who may or may not accept the investigators' reports as accurate. For instance, it is not unusual for a main character to be writing from within the walls of an insane asylum, where he has been placed by the authorities because his story is too difficult to believe.
Because Lovecraft's narrators are trustworthy to the reader--this is a result of Lovecraft's careful, precise attention to realism and current scientific theory and evidence--the narrative they offer is believable as well. This corresponds in True Detective to Cohle's notebooks and extensive documentation of the cult's activities, culminating in the damning videotape of Marie Fontenot's rape and murder, shown to Hart in episode 7, "After You're Gone." The modern-day correspondence to realism and scientific evidence is video, and that level of evidence is what Hart requires to understand fully the extent of the cult's activities. The simultaneity of eldritch horror and human reality is Lovecraft's specialty, and its translation in True Detective follows both detective fiction and weird fiction conventions.
In True Detective, just as in Lovecraft's fiction, the characters' actions are driven by an occult text. Once Cohle discovers Dora Lange's diary with its passages on the Yellow King, those elements of the narrative become codified. Because they are in text, they are quantifiable and believable. (Had Dora Lange or another cultist/victim showed up chanting "Cassilda's Song" from The King in Yellow, that character would have been written off as more than unreliable--s/he would have been unbelievable.) Lange's diary assumes its place among the other investigative documents, not the least of which is the series of police reports marked "filed in error" as part of the cover-up. Part of the reason why the written diary assumes its importance is that it can be referred to multiple times without risk of narrative change or error. If a person were relating Lange's experience, that recollection might change under different circumstances. A printed text cannot do that--unless, of course, it is a palimpsest.
Fortunately, the epistolary narrative is not the only one available to us and to the detectives. There's also a poetic narrative that surfaces during interrogations, especially the too-brief interrogation of Reggie LeDoux:
I know what happens next. I saw you in my dream. You're in Carcosa now with me. He sees you. You'll do this again. Time is a flat circle. The sun sinks into a twin black star. (episode 5, "The Secret Fate of All Life")
Cohle's explication of LeDoux's statement is equally poetic:
In eternity, where there is no time, nothing can grow, nothing can become. Nothing changes. So Death created time to grow the things that it would kill. And you are reborn, but into the same life that you've always been born into. I mean, how many times have we had this conversation, detectives? Who knows? I mean, you can't remember your lives. You can't change your lives. And that is the terrible and the secret fate of all life. You're trapped. Like a nightmare you keep waking up into. (episode 5, "The Secret Fate of All Life")But there is a false narrative, originating from Gilbough and Patania, that wars against the actual narrative, witnessed by Cohle during the flashback when he investigates the school. The police officers are following the false narrative because they've established the reality as they understand it. Cohle doesn't assume anything and is thus open to new narrative.
The entire investigation past Dora Lange's murder is spurred by philosophy, not police procedural tactics. Once Cohle hears Guy Francis's claim that the murderer is still free, Cohle's philosophical position as an artist-investigator will not permit him to let the claim go uninvestigated. As an artist, his guiding point is to explicate the truth, just as it is for himself as an investigator. It is philosophically impossible for Cohle to avoid continuing the investigation into the cult and its activities.
The cult activity itself leads the investigators to follow the path of the cultists, including the tangential paths of their sacrifices. In the world of True Detective, everything is sacrificial. Hart sacrifices his family for the sake of his adulterous relationships. Cohle sacrifices himself for the sake of his clarity as an artistic, semi-monastic investigator. Gilbough and Patania, on the other hand, sacrifice nothing, instead seeking to gain as the result of their investigation. That is precisely why they cannot and will not be able to find the true murderer--they cannot be the true detectives.
The problem is that, for the true detective, discovering the truth means teetering on the edge of sanity. Lovecraft's investigators, regardless of what occupation they might have had before entering the narrative, rarely escape confinement, peril, or death. The truth is simply too difficult to face. Similarly in True Detective, the truth about the extent of the cult's activities is limited to what can be safely exposed after Errol Childress's death. Nothing about his family connections can be revealed without dire consequence to the very social and political structures of south Louisiana.
The fact that there are circumstances that are not believable unless the narrator is an initiate or an investigator is yet another link between True Detective and Lovecraft. The very fabric of Lovecraft's narratives are circumstances that exist outside--and sometimes well outside--the normal experiences of the general populace. It's that general populace who must be spared from the knowledge that murders and worse occur because of fanaticism. It's the concealed madmen within the general cloak of normalcy that is the final betrayal.