  Studying a certain sub class of higher order Horndeski (scalar-tensor)
theories we discuss a method discovered recently permitting analytic black hole
solutions with a non trivial and regular scalar field. One of the solutions
found has de Sitter asymptotics and self tunes the bulk cosmological constant.
Using the aforementioned method we find and analyse new black hole solutions
which can also have de Sitter asymptotics. By looking at small deviations of
the integration constant responsible for self tuning we discuss the robustness
of the self tuning mechanism. We find that neighboring solutions to the one
previously found present also self tuning properties with unaltered effective
cosmological constant.
