In this paper, we study thermodynamics, thermal fluctuations, phase transitions and the charged anti-de Sitter black hole surrounded by perfect fluid dark matter. Large black holes are shown to be stable when subject to thermal fluctuations, and we begin by exploring how these fluctuations affect the uncorrected thermodynamic quantities of entropy, Helmholtz free energy, Gibbs free energy, enthalpy specific heat, and phase transition stability. We also discuss null geodesics and the radius of the photon sphere for the charged AdS BH and use the radius of a photon sphere to calculate the Lyapunov exponent and angular velocity. Exceptionally, we test the effects of various parameters of a black hole graphically by observing the existence of the correction parameter and the coupling parameter, which reveal the behavior of corrected thermodynamic quantities. Lastly, we see how the system is stable (under the effects of the dark matter parameter) by figuring out the specific heat and Hawking temperature, which are both related to entropy.

This paper deals with the thermodynamical properties of the black hole formulated in Einstein's theory of relativity associated with a nonlinear electromagnetic field. The transition of the black hole is analyzed using the mass, electric charge, coupling constant, and cosmological constant. We examine the thermodynamical aspects of exact black hole solutions to compute the black hole mass, temperature, entropy, Gibbs free energy, specific heat, and critical exponents in the phase space. Further, we study the stability of the black hole solution using the specific heat and Gibbs free energy. We examine the first and second phase changes and show a P-V criticality, which is similar to the van der Waals phase change. We also examine the equation of the state and the critical exponents.

We study the impact of thermal fluctuations on the thermodynamics, quasi-normal modes, and phase transitions of an anti-de Sitter Euler-Heisenberg black hole (BH) with a nonlinear electrodynamic field. An anti-de Sitter Euler-Heisenberg BH with a nonlinear electrodynamic field is composed of four parameters: the mass, electric charge, cosmological constant, and Euler-Heisenberg parameter. We calculate thermodynamic variables such as Hawking temperature, entropy, volume, and specific heat, which comply with the first law of thermodynamics. First, we use this BH to determine the thermodynamics and thermal fluctuations with the Euler-Heisenberg parameter to distinguish their effect on uncorrected and corrected thermodynamical quantities. We derive the expression for corrected entropy to study the impact of thermal fluctuation with simple logarithmic corrections on unmodified thermodynamical potentials, including Helmholtz energy, pressure, Gibbs free energy, and enthalpy. The Euler-Heisenberg parameter improves BH stability at large radii. Second, we analyze the local stability of the proposed BH, and the phase shifts of the BH are also investigated using temperature and specific heat. When there is a decrease in charge and an increase in and α, the temperature shifts from an unstable region to a stable one. Similarly, increases in local stability are observed with each of these parameters. Third, we use null geodesics to deal with the effects of nonlinear electrodynamics on the quasi-normal modes of the Euler-Heisenberg anti-de Sitter BH. The null geodesics provide the angular velocity and Lyapunov exponent of the photon sphere, which are the same as the real and imaginary parts of the quasi-normal modes in the eikonal limit.

In this article, motivated by the high interest in Euler–Heisenberg nonlinear electrodynamic theory, we investigated several thermal aspects of the charged Euler–Heisenberg black hole surrounded by perfect fluid dark matter. To ensure the validity of the first law of thermodynamics, we measured the conserved thermal quantities. Furthermore, the local and global thermal stability of the charged Euler–Heisenberg black hole surrounded by perfect fluid dark matter is also discussed in canonical and grand canonical ensembles and reveals how the coupling parameter affects the stability regions. We also examined the extended second-order phase transition via P–V criticality. Finally, we extend our study to the optical features of the charged Euler–Heisenberg black hole surrounded by perfect fluid dark matter, such as the black hole shadow and energy emission rate and observe the impact of the coupling parameter.