  This work is devoted to studying the magnon-magnon interaction effect in a
two-dimensional checkerboard ferromagnet with the Dzyaloshinskii-Moriya
interaction. By means of the first-order Green function formalism, the
influence of magnon-magnon interaction on the magnon band topology is analyzed.
In order to verify that the gap-closing phenomenon is a signature for the
topological phase transitions of the checkerboard ferromagnet, we display that
the Chern numbers of renormalized magnon bands are distinct above and below the
critical temperature. Our results show that the checkerboard ferromagnet
possesses two topological phases and its topological phase can be controlled
either by the temperature or applied magnetic field due to magnon-magnon
interactions. Interestingly, we find that the topological phase transition
occurs twice with the increase of the temperature, which is different from the
result of the honeycomb ferromagnet.
