  We look for the existence of asymptotically flat simple compactifications of
the form $M_{D-p}\times T^{p}$ in $D$-dimensional gravity theories with higher
powers of the curvature. Assuming the manifold $M_{D-p}$ to be spherically
symmetric, it is shown that the Einstein-Gauss-Bonnet theory admits this class
of solutions only for the pure Einstein-Hilbert or Gauss-Bonnet Lagrangians,
but not for an arbitrary linear combination of them. Once these special cases
have been selected, the requirement of spherical symmetry is no longer relevant
since actually any solution of the pure Einstein or pure Gauss-Bonnet theories
can then be toroidally extended to higher dimensions. Depending on $p$ and the
spacetime dimension, the metric on $M_{D-p}$ may describe a black hole or a
spacetime with a conical singularity, so that the whole spacetime describes a
black or a cosmic $p$-brane, respectively. For the purely Gauss-Bonnet theory
it is shown that, if $M_{D-p}$ is four-dimensional, a new exotic class of black
hole solutions exists, for which spherical symmetry can be relaxed.
  Under the same assumptions, it is also shown that simple compactifications
acquire a similar structure for a wide class of theories among the Lovelock
family which accepts this toroidal extension.
  The thermodynamics of black $p$-branes is also discussed, and it is shown
that a thermodynamical analogue of the Gregory-Laflamme transition always
occurs regardless the spacetime dimension or the theory considered, hence not
only for General Relativity.
  Relaxing the asymptotically flat behavior, it is also shown that exact black
brane solutions exist within a very special class of Lovelock theories.
