  In this work we show that Einstein gravity in four dimensions can be
consistently obtained from the compactification of a generic higher curvature
Lovelock theory in dimension $D=4+p$, being $p\geq1$. The compactification is
performed on a direct product space
$\mathcal{M}_D=\mathcal{M}_4\times\mathcal{K}^p$, where $\mathcal{K}^p$ is a
Euclidean internal manifold of constant curvature. The process is carried out
in such a way that no fine tuning between the coupling constants is needed. The
compactification requires to dress the internal manifold with the flux of
suitable $p$-forms whose field strengths are proportional to the volume form of
the internal space. We explicitly compactify Einstein-Gauss-Bonnet theory from
dimension six to Einstein theory in dimension four and sketch out a similar
procedure for this compactification to take place starting from dimension five.
Several black string/p-branes solutions are constructed, among which, a five
dimensional asymptotically flat black string composed of a Schwarzschild black
hole on the brane is particularly interesting. Finally, the thermodynamic of
the solutions is described and we find that the consistent compactification
modifies the entropy by including a constant term, which may induce a departure
from the usual behavior of the Hawking-Page phase transition. New scenarios are
possible in which large black holes dominate the canonical ensamble for all
temperatures above the minimal value.
