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 Padé 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
α and β, 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 α decrease, while other GUP parameter β 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.