Researchers have investigated static spherically symmetric vacuum solutions within the framework of $f(R)$ gravity in higher dimensions. Their work initially focused on the five-dimensional Starobinsky model, described by the action $f(R) = R + \alpha R^2 - 2\Lambda$. By enforcing the ghost-free stability criterion ($f'(R) > 0$) on constant scalar curvature spacetimes, they demonstrated that a stable effective cosmological constant cannot be dynamically generated solely from pure $R^2$ geometric corrections in five dimensions; its existence is inextricably tied to a bare cosmological constant.

Generalizing this analysis to an arbitrary number of dimensions $D$ and single-term curvature corrections of the form $f(R) = R + \alpha R^n$ with a vanishing bare cosmological constant, the scientists derived a universal stability bound: $n > D/2$. This threshold is required for the existence of stable emergent vacua. This finding suggests that, under these conditions, higher-order corrections in $f(R)$ gravity must be sufficiently pronounced to ensure stability.

However, the study also reveals that expanding the gravitational action to a multi-term polynomial hierarchy can circumvent this strict limitation. By including curvature corrections up to $\mathcal{O}(R^3)$, the extended geometric degrees of freedom simultaneously satisfy the trace constraint and the stability criterion. Furthermore, the exact parameter space boundaries have been established that ensure not only asymptotic vacuum stability but also strict global stability ($f'(R) > 0$ for all $R$), allowing for the dynamical generation of exact, globally ghost-free vacuum spacetimes in $D \ge 5$ purely from higher-order geometric terms.