  In this paper, we investigate the topological numbers of the four-dimensional
Schwarzschild black hole, $d$-dimensional Reissner-Nordstr\"om (RN) black hole,
$d$-dimensional singly rotating Kerr black hole and five-dimensional
Gauss-Bonnet black hole via the R\'enyi statistics. We find that the
topological number calculated via the R\'enyi statistics is different from that
obtained from the Gibbs-Boltzmann (GB) statistics. However, what is interesting
is that the topological classifications of different black holes are consistent
in both the R\'enyi and GB statistics: the four-dimensional RN black hole,
four-dimensional and five-dimensional singly rotating Kerr black holes,
five-dimensional charged and uncharged Gauss-Bonnet black holes belong to the
same kind of topological class, and the four-dimensional Schwarzschild black
hole and $d(>5)$-dimensional singly rotating Kerr black holes belong to another
kind of topological class. In addition, our results suggest that the
topological numbers calculated via the R\'enyi statistics in asymptotically
flat spacetime background are equal to those calculated from the standard GB
statistics in asymptotically AdS spacetime background, which provides more
evidence for the connection between the nonextensivity of the R\'enyi parameter
$\lambda$ and the cosmological constant $\Lambda$.
