Gravitation theories selected by requiring that they have a unique anti–de Sitter vacuum with a fixed cosmological constant are studied. For a given dimension d, the Lagrangians under consideration are labeled by an integer $k=1,2,\dots ,\left[\right(d-1)/2].$ Black holes for each d and k are found and are used to rank these theories. A minimum possible size for a localized electrically charged source is predicted in the whole set of theories, except general relativity. It is found that the thermodynamic behavior falls into two classes: If $d-2k=1,$ these solutions resemble the three dimensional black hole; otherwise, their behavior is similar to the Schwarzschild-${\mathrm{AdS}}_4$ geometry.

• Received 29 March 2000

DOI:https://doi.org/10.1103/PhysRevD.62.084013