Boson entropy in the NUT-Kerr-Newman black holes in the background of de Sitter space-timet

It is well known that the Euler characteristic number of most black holes is 2 (or 0), and that only NUT-Kerr-Newman black holes' Euler characteristic number is over 2. Therefore, it is significant to calculate the entropy in the NUT-Kerr-Newman black holes. In this paper, we calculated the entropy of boson field in the NUT-Kerr-Newman black holes in the background of the de Sitter space-time by using the improved membrane model established with the brick-wall method. The results show that if the cut-off is properly chosen, the entropy in the black holes still satisfies the Bekenstein-Hawking area law.