The word "pertinacity", in the excerpt from "The Black Cat", by Edgar Allan Poe, most likely means the following :
Obstinacy.
The cat was obstinate in following the narrator around his house. The more the narrator would hate it, the more the animal would insist on doing it. It was as if the cat were some sort of entity who would always appear wherever the speaker was, so he could be reminded of his intense aversion. Pluto (the cat's name) always got in the way of the narrator, such as in the passage where the narrator kills his wife and also in the conclusion of the story.