Statistical entropy of a stationary dilaton black hole from the Cardy formula

With Carlip's boundary conditions, a standard Virasoro subalgebra with a corresponding central charge for a stationary dilaton black hole obtained in the low-energy effective field theory describing string is constructed at a Killing horizon. The statistical entropy of a stationary dilaton black hole yielded by the standard Cardy formula agrees with its Bekenstein-Hawking entropy only if we take the period T of function v as the periodicity of the Euclidean black hole. On the other hand, if we consider a first-order quantum correction then the entropy contains a logarithmic term with a factor -1/2, which is different from that of Kaul and Majumdar, -3/2. We also show that the discrepancy is not just for the dilaton black hole, but for any one whose corresponding central charge takes the form c/12 = (A(H)/8 piG)(2 pi /kT).