Some very interesting solutions of the field equations of Einstein's general theory of relativity have been constructed in the framework of nonlinear electrodynamics. In particular, magnetically charged black hole solutions in the framework of exponential nonlinear electrodynamics have been obtained in general relativity in Kruglov (2017) [13]. Using this approach a magnetically charged non-singular black hole spacetime in the framework of exponential electrodynamics in some modified theory of gravity has been constructed in this letter. The metric describing asymptotic non-singular magnetized black hole is worked out in terms of the parameter of our model. When this parameter vanishes our solution reduces to the above mentioned black hole solution. Thermodynamics of the resulting solution is also discussed by calculating the Hawking temperature and heat capacity when the magnetic charge is constant. We also find out the point where the first order phase transition induced by temperature changes takes place. The quantum radiations from this black hole are also discussed and the mathematical expression for the rate of energy flux of these radiations has been obtained. arXiv:1904.05727v1 [gr-qc]

In this paper, a large family of topological black hole solutions of dimensionally continued gravity are derived. The action of Lovelock gravity is coupled to the exponential electrodynamics and the equations of motion are solved in the presence of a pure magnetic source. We work out the metric functions in terms of the parameter β of exponential electrodynamics, and magnetic charge. Further, we couple Lovelock gravity to power-Yang-Mills theory and construct black holes, in diverse dimensions, having Yang-Mills magnetic charge. We also discuss the asymptotic bahaviour of metric functions and curvature invariants at the origin for both the models. The thermodynamics of resulting magnetized black hole solutions in the framework of two different models is also studied. The thermodynamical quantities like Hawking temperature, entropy and specific heat capacity at constant charge are found and we show that the resulting quantities satisfy the first law of black hole thermodynamics. We also study the magnetized hairy black holes of dimensionally continued gravity.

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Lovelock gravity containing dimensionally continued Euler densities can be considered as a natural extension of Einstein’s theory of gravity in higher dimensions such that the associated differential equations of motion are still second order. In this paper, the Lovelock gravity is coupled with a scalar field and dimensionally continued hairy black holes in the presence of quintessential matter and cloud of strings are analyzed. Thermodynamics of these black holes is discussed as well. It is shown that the thermodynamic quantities satisfy the generalized first law. The generalized Smarr’s relation is also derived and thermodynamic stability checked. Finally, black holes of general Lovelock-scalar theory are also briefly discussed.