Generalized (2+1) dimensional black hole by Noether symmetry

We use the Noether symmetry approach to find f (R) theory of (2 + 1) dimensional gravity and (2 + 1) dimensional black hole solution consistent with this f (R) gravity and the associated symmetry. We obtain f (R) = D1R(n/n+1)(R/K)(1/n) + D2R + D-3, where the constant term D-3 plays no dynamical role. Then, we find general spherically symmetric solution for this f (R) gravity which is potentially capable of being as a black hole. Moreover, in the special case D-1 = 0, D-2 = 1, namely f (R) = R + D-3, we obtain a generalized BTZ black hole which, other than common conserved charges m and J, contains a new conserved charge Q. It is shown that this conserved charge corresponds to the freedom in the choice of the constant term D-3 and represents symmetry of the action under the transformation R -> R' = R + D-3 along the killing vector partial derivative(R). The ordinary BTZ black hole is obtained as the special case where D-3 is fixed to be proportional to the infinitesimal cosmological constant and consequently the symmetry is broken via Q = 0. We study the thermodynamics of the generalized BTZ black hole and show that its entropy can be described by the Cardy-Verlinde formula.