Thermodynamic properties of the Kerr Black-hole in non-linear electrodynamics with cosmological constant

We study thermodynamic properties, in particular the Temperature~(T), Angular velocity~(${\mathrm{\Omega }}_h$) and Entropy~(S) of the of magnetically charged slowly rotating (with rotation parameter $a\text{lsim}0.10$) Kerr black-hole(BH) with the inclusion of cosmological constant ($\mathrm{\Lambda }$) in the background of nonlinear electrodynamics (NLED). At first we calculated the nonlinear electromagnetic magnetic charged density ${\rho }_{NLED}$$(r)$ which is needed to calculate the magnetic mass of the slowly rotating Kerr-BH. We showed the mass profile $M(r)$ of the BH for different combinations of magnetic charges~($q_m$) and non-linear parameter ($\beta$) presence in the Lagrangian density. We found that $M(r)$ attains a plateau for values of $r$ close to the the cosmological length~($L$), where $L^2$= $\pm \frac{\mathrm{\Lambda }}{3}$, irrespective of the combinations of $q_m$ and $\beta$. The $\pm$ sign corresponds to the de-Sitter(dS) and Anti-de-Sitter(AdS) respectively. Afterwards we showed the allowed parameter spaces in $a$-$M$ plane using sharkfin diagram for different values of $L$ with positive values of the horizon function, $\mathrm{\Delta }(r)$, and explain the extremal criterion and asymptotic limit. We showed the values of $r$ where the horizon function of the quadratic polynomials becomes zero and called them as the inner(Cauchy), outer(Event) and large cosmological horizons with different values of $a$. We showed that the horizon structure depends on $a$, $L$ and the mass profile $M(r)$. Finally, we tabulated the numerical values of three thermodynamic parameter, i.e., $T$, ${\mathrm{\Omega }}_h$ and $S$ at those horizons surfaces. Our results demonstrate that NLED with cosmological constant significantly modifies both the internal structure and thermodynamic properties of Kerr-BH.