  We investigate the black hole solutions in the $R^2$-gravity, where the
action contains the square of the curvature. In case that the action does not
contain the square of the Riemann tensor and in case that the $R^2$-terms are
the Gauss-Bonnet (GB) combination, we find exact solutions. We investigate the
thermodynamics of these theories and find the Hawking-Page like phase
transition, which is the phase transition between the black hole (BH) spacetime
and the pure anti-deSitter (AdS) spacetime. From the viewpoint of the AdS/CFT
correspondence, such a phase transition may correspond to thermal transition of
dual CFT.
  An interesting feature of $R^2$-gravity is the possibility of the negative
(or zero) dS (or AdS) BH entropy, which depends on the parameters of the
$R^2$-terms. We speculate that the appearence of negative entropy may indicate
a new type instability where a transition between dS (AdS) BH with negative
entropy and AdS (dS) BH with positive entropy occurs.
  We also apply the GB gravity to the brane cosmology, where the brane moves in
the bulk AdS BH spacetime. By investigating the FRW-like equation, which
describes the motion of the brane, we find the behavior of the matter on the
brane. When the radius of the brane is large, the matter fields behave as CFT
but when the radius is small, the brane universe behaves as the universe with
dust or curvature dominant universe, depending on the parameters.
