  By reintroducing Lorentz invariance in canonical loop quantum gravity, we
define a geometrical notion of temperature for quantum isolated horizons. This
is done by demanding that the horizon state satisfying the boundary conditions
be a Kubo-Martin-Schwinger state. The exact formula for the temperature can be
derived by imposing the reality conditions in the form of the linear simplicity
constraints for an imaginary Barbero-Immirzi parameter. Thus, our analysis
reveals the connection between the analytic continuation to the Ashtekar
self-dual variables and the thermality of the horizon. The horizon thermal
equilibrium state can then be used to compute both the entanglement and the
Boltzmann entropies. We show that the two provide the same finite answer, which
allows us to recover the Bekenstein-Hawking formula in the semi-classical
limit. In this way, we shed new light on the microscopic origin of black hole
entropy by revealing the equivalence between the near-horizon degrees of
freedom entanglement proposal and the state-counting interpretation. The
connection with the Connes-Rovelli thermal time proposal for a general
relativistic statistical mechanics is worked out.
