In this paper, by utilizing the rainbow functions that were proposed by Amelino-Camelia et al., the information flux of rainbow Schwarzschild black hole and the sparsity of Hawking radiation in rainbow gravity are explored. The results show that the rainbow gravity has a very significant effect on the information flux. When the mass of rainbow Schwarzschild black hole approaches to the order of Planck scale, the Bekenstein entropy loss per emitted quanta in terms of the mass of Schwarzschild black hole reduces to zero. Furthermore, we also find the sparsity of Hawking radiation in rainbow gravity is no longer a constant; instead, it monotonically decreases as the mass of black hole decrease. At the final stages of evaporation, the modified sparsity becomes infinity, which indicates the effect of quantum gravity stops Hawking radiation and leads to remnant.
In this work, we construct a new kind of rainbow functions, which has generalized uncertainty principle parameter. Then, we investigate modified thermodynamic quantities and phase transition of rainbow Schwarzschild black hole by employing this new kind of rainbow functions. Our results demonstrate that the rainbow gravity and generalized uncertainty principle have a great effect on the picture of Hawking radiation. They prevent black holes from total evaporation and cause a remnant. In addition, after analyzing the modified local thermodynamic quantities, we find that the effect of rainbow gravity and the generalized uncertainty principle lead to one first-order phase transition, two second-order phase transitions, and two Hawking-Page-type phase transitions in the thermodynamic system of rainbow Schwarzschild black hole.
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