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We look for the existence of asymptotically flat simple compactifications ofthe form $M_{D-p}\times T^{p}$ in $D$-dimensional gravity theories with higherpowers of the curvature. Assuming the manifold $M_{D-p}$ to be sphericallysymmetric, it is shown that the Einstein-Gauss-Bonnet theory admits this classof solutions only for the pure Einstein-Hilbert or Gauss-Bonnet Lagrangians,but not for an arbitrary linear combination of them. Once these special caseshave been selected, the requirement of spherical symmetry is no longer relevantsince actually any solution of the pure Einstein or pure Gauss-Bonnet theoriescan then be toroidally extended to higher dimensions. Depending on $p$ and thespacetime dimension, the metric on $M_{D-p}$ may describe a black hole or aspacetime with a conical singularity, so that the whole spacetime describes ablack or a cosmic $p$-brane, respectively. For the purely Gauss-Bonnet theoryit is shown that, if $M_{D-p}$ is four-dimensional, a new exotic class of blackhole solutions exists, for which spherical symmetry can be relaxed.  Under the same assumptions, it is also shown that simple compactificationsacquire a similar structure for a wide class of theories among the Lovelockfamily which accepts this toroidal extension.  The thermodynamics of black $p$-branes is also discussed, and it is shownthat a thermodynamical analogue of the Gregory-Laflamme transition alwaysoccurs regardless the spacetime dimension or the theory considered, hence notonly for General Relativity.  Relaxing the asymptotically flat behavior, it is also shown that exact blackbrane solutions exist within a very special class of Lovelock theories.Comment: 30 pages, no figures, few typos fixed, references added, final  version for JHE

Simple compactifications and black p-branes in Gauss-Bonnet and Lovelock theories