Quadratic scale-invariant gravity non minimally coupled to a scalar field
    provides a competitive model for inflation, characterized by the transition
    from an unstable to a stable fixed point, both characterized by constant scalar
    field configurations. We provide a complementary analysis of the same model in
    the static, spherically symmetric setting, obtaining two Schwarzschild-de
    Sitter solutions in parallel to the fixed points existing in the cosmological
    scenario. The stability of such solutions is thoroughly investigated from two
    different perspectives. First, we perform a semi-classical analysis based on
    the Euclidean path integral formulation. By studying the difference between the
    Euclidean on-shell actions evaluated on both solutions, we prove that the
    unstable one has a meta-stable character and is spontaneously decaying into the
    stable fixed point which is always favoured. Then, the system is further
    studied at the classical level by the analysis of linear perturbations. In
    particular, we provide both analytical and numerical results for the late-time
    behavior of the perturbations, confirming the stable and unstable character of
    the two solutions.

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