Quantum-corrected entropy for (1 + 1)-dimensional gravity revisited

In this paper, we examine a generic theory of (1 + 1)-dimensional gravity with coupling to a scalar field. Special attention is paid to a class of models that have a power-law form of dilaton potential and can admit black-hole solutions. The study focuses on the formulation of a Lorentzian partition function. Extending a four-dimensional treatment by Makela and Repo, we incorporate the principles of Hamiltonian thermodynamics (as well as black-hole spectroscopy) and find that the partition function can be expressed in a calculable form. We then go on to extract the black-hole entropy, including the leading-order quantum correction. As anticipated, this correction can be expressed as the logarithm of the classical entropy. Interestingly, the prefactor for this logarithmic correction disagrees, in both magnitude and sign, with the findings from a prior study (on the same model). We comment on this discrepancy and provide a possible rationalization