Black hole stability has seen numerous important developments in the past decade. In my talk, I will focus on the nonlinear stability of the slowly-rotating Kerr-de Sitter family. Nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by Hintz and Vasy in 2016 using microlocal techniques. In my talk, I will present a new proof of the nonlinear stability of slowly-rotating Kerr-de Sitter spacetimes that utilizes the vectorfield method to uncover the high-frequency spectral gap in Kerr-de Sitter, connecting the physical-space and frequency-space approaches. The proof also uses a bootstrap scheme to prove nonlinear stability, avoiding Nash-Moser.