Thermodynamics and null-geodesic of the Kerr-Newman black hole surrounded by quintessence and a cloud of string

In this paper, we study the effect of the modified dispersion relation (MDR) on the thermodynamics of the Kerr-Newman black hole surrounded by quintessence and a cloud of string. The thermodynamic properties of the Kerr-Newman black hole are shown to rely not only on the black hole’s properties but also on the parameters associated with the modified dispersion relation, quintessence, and the cloud string. Additionally, the equation of state is impacted by these parameters. The remnant and the stability of the black hole are also discussed under the correction of MDR, quintessence, and a cloud of string. In addition, the null geodesic structure of the spacetime is studied using the Hamilton–Jacobi formalism. The effective potential for photon motion is obtained, and the radii of the prograde and retrograde circular photon orbits are determined. The instability of these circular photon orbits is further characterized by the Lyapunov exponent.