  In this paper we construct compactifications of generic, higher curvature
Lovelock theories of gravity over direct product spaces of the type
$\mathcal{M}_D=\mathcal{M}_d \times \mathcal{S}^p $, with $D=d+p$ and $d\ge5$,
where $\mathcal{S}^p$ represents an internal, Euclidean manifold of positive
constant curvature. We show that this can be accomplished by including suitable
non-minimally coupled $p-1$-form fields with a field strength proportional to
the volume form of the internal space. We provide explicit details of this
constructions for the Einstein-Gauss-Bonnet theory in $d+2$ and $d+3$
dimensions by using one and two-form fundamental fields, and provide as well
the formulae that allows to construct the same family of compactification in
any Lovelock theory from dimension $d+p$ to dimension $d$. These fluxed
compactifications lead to an effective Lovelock theory on the compactfied
manifold, allowing therefore to find, in the Einstein-Gauss-Bonnet case, black
holes in the Boulware-Deser family.
