Rotating Lifshitz-like black holes in F(R) gravity

One of the alternative theories of gravitation with a possible UV completion of general relativity is Horava-Lifshitz gravity. Regarding a particular class of pure F(R) gravity in three dimensions, we obtain an analytical rotating Lifshitz-like black hole solution. We first investigate some geometrical properties of the obtained solution that reduces to a charged rotating Banados-Teitelboim-Zanelli black hole in a special limit. Then, we study the optical features of such a black hole, like the photon orbit and the energy emission rate, and discuss how electric charge, angular momentum, and exponents affect them. To have acceptable optical behavior, we should apply some constraints on the exponents. We continue our investigation by studying the thermodynamic behavior of solutions in the extended phase space and explore the validity of the first law of thermodynamics as well as local thermal stability using heat capacity. Evaluating the existence of van der Waals-like phase transition, we obtain critical quantities and show how they change under the variation of black hole parameters. Finally, we construct a holographic heat engine of such a black hole and obtain its efficiency in a cycle. Comparing the obtained results with the well-know Carnot heat engine efficiency, we examine the second law of thermodynamics.