  The superradiant growth of massive vector fields in rotating black hole
spacetimes has garnered significant attention in recent literature. However,
the majority of these studies overlook the influence of a cosmological
constant, which likely constitutes the primary energy content of our universe.
In this paper, we extend recent research by incorporating a cosmological
constant into the Einstein+Proca system and numerically evolving the resulting
equations of motion. Utilizing the newly released GRBoondi numerical relativity
code, designed specifically for the numerical evolution of (generalized) Proca
fields, we discover that parameters causing a growing instability in the
$\Lambda = 0$ scenario transition to a decaying state when $\Lambda > 0$. This
results in a more intriguing phenomenology. These simulations pave the way for
future full Einstein+Proca simulations to explore the secular decay of the
resultant cloud from gravitational emission.
