We study black hole solutions in extended gravities with higher-order curvature terms, including conformal and Einstein-Weyl gravities. In addition to the usual anti-de Sitter (AdS) vacuum, the theories admit Lifshitz and Schrödinger vacua. The AdS black hole in conformal gravity contains an additional parameter over and above the mass, which may be interpreted as a massive spin-2 hair. By considering the first law of thermodynamics, we find that it is necessary to introduce an associated additional intensive/extensive pair of thermodynamic quantities. We also obtain new Lifshitz black holes in conformal gravity and study their thermodynamics. We use a numerical approach to demonstrate that AdS black holes beyond the Schwarzschild-AdS solution exist in Einstein-Weyl gravity. We also demonstrate the existence of asymptotically Lifshitz black holes in Einstein-Weyl gravity. The Lifshitz black holes arise at the boundary of the parameter ranges for the AdS black holes. Outside the range, the solutions develop naked singularities. The asymptotically AdS and Lifshitz black holes provide an interesting phase transition, in the corresponding boundary field theory, from a relativistic Lorentzian system to a nonrelativistic Lifshitz system.

Higher-order curvature corrections involving the conformally invariant Weyl-squared action have played a role in two recent investigations of four-dimensional gravity: in critical gravity, where they are added to the standard cosmological Einstein-Hilbert action with a coefficient tuned to make the massive ghostlike spin-2 excitations massless, and in a pure Weyl-squared action considered by Maldacena, where the massive spin-2 modes are removed by the imposition of boundary conditions. We exhibit the connections between the two approaches, and we also generalize critical gravity to a wider class of Weyl-squared modifications to cosmological Einstein gravity where one can eliminate the massive ghostlike spin-2 modes by means of boundary conditions. The cosmological constant plays a crucial role in the discussion, since there is then a “window” of negative mass-squared spin-2 modes around ${\mathrm{AdS}}_4$ that are not tachyonic. We also construct analogous conformal and nonconformal gravities in six dimensions.

We construct supersymmetric completions of various curvature squared terms in five dimensional supergravity with eight supercharges. Adopting the dilaton Weyl multiplet, we obtain the minimal off-shell supersymmetric Ricci scalar squared as well as all vector multiplets coupled curvature squared invariants. Since the minimal off-shell super- symmetric Riemann tensor squared and Gauss-Bonnet combination in the dilaton Weyl multiplet have been obtained before, both the minimal off-shell and the vector multiplets coupled curvature squared invariants in the dilaton Weyl multiplet are complete. We also constructed an off-shell Ricci scalar squared invariant utilizing the standard Weyl multiplet. The supersymmetric Ricci scalar squared in the standard Weyl multiplet is coupled to n number of vector multiplets by construction, and it deforms the very special geometry. We found that in the supersymmetric AdS ₅ vacuum, the very special geometry defined on the moduli space is modified in a simple way. Finally, we study the magnetic string and electric black hole solutions in the presence of supersymmetric Ricci scalar squared.