We present an exact static, spherically symmetric black hole solution to the third-order Lovelock gravity with a string cloud background in seven dimensions for the special case when the second-and third-order Lovelock coefficients are related viaα 2 2 = 3α 3 (≡ α 2 ). Further, we examine thermodynamic properties of this black hole to obtain exact expressions for mass, temperature, heat capacity and entropy, and also perform the thermodynamic stability analysis. We see that a string cloud background has a profound influence on the horizon structure, thermodynamic properties, and the stability of black holes. Interestingly the entropy of the black hole is unaffected due to the string cloud background. However, the critical solution for thermodynamic stability is affected by the string cloud background.

Lovelock gravity consisting of the dimensionally continued Euler densities is a natural generalization of general relativity to higher dimensions such that equations of motion are still second order, and the theory is free of ghosts. A scalar field with a positive potential that yields an accelerating universe has been termed quintessence. We present exact black hole solutions in D-dimensional Lovelock gravity surrounded by quintessence matter and also perform a detailed thermodynamical study. Further, we find that the mass, entropy and temperature of the black hole are corrected due to the quintessence background. In particular, we find that a phase transition occurs with a divergence of the heat capacity at the critical horizon radius, and that specific heat becomes positive for r h < r c allowing the black hole to become thermodynamically stable.

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Recently, several methods have been proposed to regularize a D→4 limit of Einstein–Gauss–Bonnet (EGB), leading to nontrivial gravitational dynamics in 4D. We present an exact nonsingular black hole solution in the 4D EGB gravity coupled to non-linear electrodynamics and analyze their thermodynamic properties to calculate precise expressions for the black hole mass, temperature, and entropy. Because of the magnetic charge, the thermodynamic quantities are corrected, and the Hawking–Page phase transition is achievable with diverges of the heat capacity at a larger critical radius r=r+C in comparison to the 5D counterpart where the temperature is maximum. Thus, we have a black hole with Cauchy and event horizons, and its evaporation leads to a thermodynamically stable extremal black hole remnant with vanishing temperature, and its size is larger than the 5D counterpart. The entropy does not satisfy the usual exact horizon Bekenstein–Hawking area law of general relativity with a logarithmic area correction term.