The work is devoted to the study of thermodynamic properties of three field-theoretical models of black holes: static and rotating, electrically and magnetically charged, with different non-linear electromagnetic fields, in various spacetime dimensions and topologies within General Relativity. The first considered model is a static, electrically charged black hole with four various types of non-linear electromagnetic field, namely, power Maxwell invariant, Born-Infeld, logarithmic and exponential in three-dimensional spacetime with a negative cosmological constant. Equations of gravitational and electromagnetic fields, and their exact analytical solutions are obtained. It is shown that the electric field in the Born-Infeld and logarithmic electrodynamics does not diverge at the origin. The Born-Infeld type fields possess similar qualitative behavior. The power Maxwell invariant field with the non-linearity parameter equal to unity corresponds to linear Maxwell electrodynamics. The conformal source of the power Maxwell invariant is investigated. In this case the electric field satisfies the inverse-square law in three dimensions, electromagnetic stress-energy tensor is traceless, and scalar curvature takes a constant value. Relations between integration constants and macroscopic parameters of black hole, namely, mass and full electric charge are established. The Hawking temperature and electric potential on the black hole horizon are calculated. The extended phase space thermodynamics where a negative cosmological constant is associated as environment pressure is investigated. The first law of thermodynamics is written for the black hole mass, which is interpreted with enthalpy. The black hole thermodynamic volume coincides with the area of a circle of the black hole horizon radius. The equation of state of the black hole is written and its trivial behavior is discovered. Also, the heat capacity at constant pressure is calculated. The second examined model is a slowly rotating, electrically charged black hole with power Maxwell invariant electromagnetic field in three-dimensional spacetime with a negative cosmological constant. Field equations and their solutions are obtained. These black hole solutions are described by two metric functions, namely diagonal and nondiagonal functions, and also two components of electromagnetic field tensor, namely the radial electric and magnetic fields. The slow rotation limit is defined by a small parameter which is related to the black hole angular momentum. For the magnetic field and non-diagonal metric function exact quadrature as well as analytical asymptotic solutions are found, these results are original. The solutions for electric and magnetic fields are obtained in a general case without the assumption of additional connections between them. Relations between integration constants and black hole parameters — its mass, electric charge and angular momentum are discussed. The third studied model is a static, electrically and magnetically charged black hole in the Born-Infeld electrodynamics of four-dimensional spacetime with spherical, planar and hyperbolic black hole horizons with a negative cosmological constant. The electromagnetic Born-Infeld lagrangian is chosen in its original form with two field invariants. Field equations and its exact analytical solutions are obtained. The black hole is described by one diagonal metric function, radial electric and angular magnetic fields. The ansatz for the electromagnetic field potential was chosen in the form which gives rise to the same form of electric field and field invariants for all types of considered horizon geometries. The electric field grows slowly near the origin for larger values of the magnetic charge for fixed the electric one and is finite. Thermodynamic behavior of the obtained solutions in the extended phase space thermodynamics are examined. The first law of black hole thermodynamics and Smarr relation for finite thermodynamic quantities with additional terms of the Born-Infeld vacuum polarization are written. The black hole equation of state is obtained and its critical behavior is investigated. A phase transition occurs only for the spherical black hole horizon, and conditions for its existence are given. The thermodynamic critical ratio is found and its original features are analyzed. Also, the heat capacity at constant pressure is obtained.
