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Stationary and Free-fall frame Kerr black hole in gravity's rainbow

Doubly special relativity (DSR) is an effective model for encoding quantum gravity in flat spacetime. To incorporate DSR into general relativity, one could use gravity's rainbow, where the spacetime background felt by a test particle would depend on its energy. In this paper, we investigate the thermodynamics of rainbow Kerr black hole in the scenario with the stationary(ST) orthonormal frame and free-fall(FF) orthonormal frame. After the rainbow metric in ST frame and FF frame is deduced, the Hamilton-Jacobi method is used to acquire the modified Hawking temperature, specific heat and corresponding the modified entropy to each scenario, then the thermodynamic properties are discussed. We find that the effects of rainbow gravity on Kerr black holes are quite model-dependent. In other words, the value of parameter $η$ and $n$ with Amelino-Camelia's proposal are crucially important and worth discussing. Specificly, with most widly accepted choice ($n=2,η>0$), the effects of rainbow gravity tend to decrease the Hawking temperature but increase the black hole entropy in ST frame, and increase the Hawking temperature but decrease the black hole entropy in FF frame conversely.