  We define the analytic continuation of the number of black hole microstates
in Loop Quantum Gravity to complex values of the Barbero-Immirzi parameter
$\gamma$. This construction deeply relies on the link between black holes and
Chern-Simons theory. Technically, the key point consists in writing the number
of microstates as an integral in the complex plane of a holomorphic function,
and to make use of complex analysis techniques to perform the analytic
continuation. Then, we study the thermodynamical properties of the
corresponding system (the black hole is viewed as a gas of indistinguishable
punctures) in the framework of the grand canonical ensemble where the energy is
defined \'a la Frodden-Gosh-Perez from the point of view of an observer located
close to the horizon. The semi-classical limit occurs at the Unruh temperature
$T_U$ associated to this local observer. When $\gamma=\pm i$, the entropy
reproduces at the semi-classical limit the area law with quantum corrections.
Furthermore, the quantum corrections are logarithmic provided that the chemical
potential is fixed to the simple value $\mu=2T_U$.
