  A metric-affine approach is employed to study higher-dimensional modified
gravity theories involving different powers and contractions of the Ricci
tensor. It is shown that the field equations are \emph{always} second-order, as
opposed to the standard metric approach, where this is only achieved for
Lagrangians of the Lovelock type. We point out that this property might have
relevant implications for the AdS/CFT correspondence in black hole scenarios.
We illustrate these aspects by considering the case of Born-Infeld gravity in
$d$ dimensions, where we work out exact solutions for electrovacuum
configurations. Our results put forward that black hole singularities in
arbitrary dimensions can be cured in a purely classical geometric scenario
governed by second-order field equations.
