It's by no means rare or unusual to see the Seven Deadly Sins used as a thematic element or trope in movies, TV, books or comics. Somehow, there are still new and fresh ways to use them so it avoids getting stale (usually). But I have noticed a very weird trend when it comes to the Seven Deadlies in anime. Two for two on the very visible uses of this theme, the characters assigned to Sloth are actively sadistic psychopaths. They are more likely to cut your face off than the Wrath characters.
The first time I saw this was in Angel Sanctuary, the lush psychological funhouse of unrestrained ideas by Kaori Yuki. Personally, I'm a big fan of the first 16 books of this series and then it peters out into the confusing and disappointing, but that's a bit unrelated to my point. Angel Sanctuary uses a concept of seven major demons that are associated with the Seven Deadlies. The demon associated with Sloth is Astaroth, Duke of Terror. We had already been introduced to Astaroth by the time we learned his association with Sloth, and I remember finding it extremely jarring.
Astaroth's domain is a massive labyrinth that is constantly reshaping itself to his will, so that he is the only one who can navigate it. He enjoys watching people get lost and die of thirst, but not as much as he enjoys capturing the unfortunate ones who fall into his territory and torturing them to death. Although there could be an element of Sloth in the fact that generally Astaroth watches this done rather than taking active part, he shows in his brief appearance in the series that he is not averse to physically raping and tearing a person limb from limb.
Astaroth is insane, possibly due to spoilery matters in his backstory. He says that he exists in a kind of constant mental pain that is only erased by the sight of fresh-flowing blood. Although his 'sloth' line when all the demons are present (and spouting their sin-related introductory lines) seems pretty straight-forward (he says, basically, "I won't help you. I'm not interested if blood doesn't flow.") it also emphasizes his extreme need for violence.
Compare this with the second example, from Kateikyo Hitman Reborn (a terrific series that I so far love all the way through). The Varia, an elite autonomous assassin squad that are Season Two antagonists, include a seven-unit elite group that are all associated (usually fairly obviously) with one of the Seven Deadly Sins. The one associated with Sloth is Belphegor, which is pretty clear since the character is named after the traditional demonic embodiment OF Sloth.
Bel is also a complete psychopath. He's a genius assassin known as Prince the Ripper, because he is apparently actually the prince of some country or other and because he likes to violently murder his victims using knives. Bel killed his own twin brother when he was very young and then explained it by saying he 'mistook him for a cockroach.'
Apparent justification for Belphegor as Sloth would lie in the fact that he left his country and his duties as prince in order to follow the path of a killer. He also, like Astaroth, will sometimes complain or even momentarily refuse to do something unless he's motivated by personal interest.
However, that is not something that is limited to the Sloth character in either AS or KHR. In Angel Sanctuary, Asmodeus (Lust) and in Reborn, Mammon (Greed) both will refuse or resist orders that help the group but that don't serve their own interests or even interest them.
So where is the distinct Slothfulness in either of these characters?
Concept 1: In Wikipedia, sloth is defined as "spiritual and/or actual apathy or laziness, putting off what God asks you to do, or not doing it or anything at all. Acedia is a Latin word, from Greek akedia, literally meaning "absence of caring"." It can also apparently refer to any waste of anything that would require work, maintenance or support to continue to exist.
The Catholic Encyclopedia says, "In general it means disinclination to labour or exertion. As a capital or deadly vice St. Thomas (II-II:35) calls it sadness in the face of some spiritual good which one has to achieve (Tristitia de bono spirituali). Father Rickaby aptly translates its Latin equivalent acedia (Gr. akedia) by saying that it means the "don't-care feeling." ... The idea of right living inspires not joy but disgust, because of its laboriousness... in this sense sloth is not a specific vice according to the teaching of St. Thomas, but rather a circumstance of all vices. Ordinarily it will not have the malice of mortal sin unless, of course, we conceive it to be so utter that because of it one is willing to bid defiance to some serious obligation."
According to these general definitions, the slothfulness of both Astaroth and Belphegor pretty much is a lump sum. If Sloth is the circumstance of all sins based upon NOT CARING or not feeling, the extreme form that would be a mortal offense might include not caring if anyone else lives or dies. So, choosing to act as a sociopath might include a form of Sloth.
I find this explanation trite and pat, however, because in both instances the motivation for violence and for cruelty seems to be a deep-seated need to do violence or to inflict pain or death upon others. In this sense, it isn't just a disdain for and disinterest in doing God's work (which would include ALL Seven Deadly Sins style characters), or work itself (which sounds more like traditional Sloth but isn't true of either of them): it is an obsession.
Option Two: Although spiritual Sloth is normally associated with a disgust for religious practices and spirituality of any kind, even a hatred of God, in one instance (that of a Duke of Hell) it's pretty much a given, and in the other, Belphegor, it's impossible to tell. Spiritual sloth COULD, on the other hand, include self-destructive qualities and self-hatred.
It also can be stretched to include mental laziness, an unwillingness to change one's viewpoint, and waste of time on pointless activities. And, although I can't find any proposition of it anywhere, it could possibly be stretched to include obsession at the cost of other obligations and other people.
If you take it that far, I can begin to see where Sloth might be hiding in Astaroth and Belphegor. Astaroth definitely has qualities of self-hatred lurking about in the black hole that is his brain. He is self-destructive, even self-mutilating (though he is exceptionally averse to and violently vengeful of others hurting him). He's set in his ways, mentally limited in scope and interest.
Belphegor doesn't display any self-destructive or self-hating qualities. In fact, he displays a high degree of pride both in his competence as a killer and his blood as a prince. He, in fact, reacts violently to any sort of attack or even teasing. He IS, on the other hand, unwilling to expand his thoughts and ideas. He could be considered to be fairly narrow-minded due to his very rigid sense of self. Mostly, his obsession with combat (and murder) to the exclusion of his former life or anything else might, just might, be considered slothful.
That's as close as I can get.