Effective four-dimensional loop quantum black hole with a cosmological constant

In this paper, we utilize the effective corrections of the $\bar{\mu}$-scheme in loop quantum black holes to obtain a 4-dimensional spherically symmetric metric with a cosmological constant. By imposing the areal gauge on the components of Ashtekar variables in the classical theory and applying the holonomy corrections, we derive the equations of motion, which can be solved to obtain the expression for the effective metric in the Painlev\'{e}-Gullstrand coordinates. Compared to the classical dS (AdS) spacetime, the LQG correction sets an upper bound on the cosmological constant as $\Lambda<\frac{3}{\gamma^2\Delta}$. The thermodynamic properties of black holes have also been calculated. We interestingly found that for a small black hole, the temperature of the LQG black hole decreases as the mass decreases, which is quite different with the classical scenario. Moreover, our result shows that a logarithmic term appeared as the leading order correction to the Beikenstein-Hawking entropy. Furthermore, the LQG corrections also introduce an extra phase transition in the black hole's heat capacity at smaller radius.