Quasinormal Modes and Hawking Radiation Sparsity of GUP corrected Black Holes in Bumblebee Gravity with Topological Defects

We have obtained the Generalized Uncertainty Principle (GUP) corrected de Sitter and anti-de Sitter black hole solutions in bumblebee gravity with a topological defect. We have calculated the scalar, electromagnetic and gravitational quasinormal modes for the both vanishing and non-vanishing effective cosmological constant using Pad\'e averaged sixth order WKB approximation method. Apart from this, the time evolutions for all three perturbations are studied, and quasinormal modes are calculated using the time domain profile. We found that the first order and second order GUP parameters $\alpha$ and $\beta$, respectively have opposite impacts on the quasinormal modes. The study also finds that the presence of a global monopole can decrease the quasinormal frequencies and the decay rate significantly. On the other hand, Lorentz symmetry violation has noticeable impacts on the quasinormal frequencies and the decay rate. We have studied the greybody factors, power spectrum and sparsity of the black hole with the vanishing effective cosmological constant for all the three perturbations. The presence of Lorentz symmetry breaking and the GUP parameter $\alpha$ decrease, while other GUP parameter $\beta$ and the presence of global monopole increase the probability of Hawking radiation to reach the spatial infinity. The presence of Lorentz violation can make the black holes less sparse, while the presence of a global monopole can increase the sparsity of the black holes. Moreover, we have seen that the black hole area quantization rule is modified by the presence of Lorentz symmetry breaking.