Statistical analysis of entropy correction from topological defects in loop black holes

In this paper we discuss the entropy of quantum black holes in the Loop quantum gravity; formalism when the number of punctures on the horizon is treated as a quantum hair, that is we compute the black hole entropy in the grand canonical (area) ensemble. The entropy is a function of both the average area and the average number of punctures and bears little resemblance to the Bekenstein-Hawking entropy. In the thermodynamic limit, both the “temperature” and the chemical potential can be shown to be functions only of the average area per puncture. At a fixed temperature, the average number of punctures becomes proportional to the average area and we recover the Bekenstein-Hawking area-entropy law to leading order provided that the Barbero-Immirzi parameter, $\gamma$, is appropriately fixed. This also relates the chemical potential to $\gamma$. We obtain a subleading correction, which differs in signature from that obtained in the microcanonical and canonical ensembles in its sign but agrees with earlier results in the grand canonical ensemble.