  Various gravity theories beyond general relativity have been rigorously
investigated in the literature such as Horndeski and degenerate higher-order
scalar-tensor (DHOST) theories. In general, numerous model parameters are
involved in such theories, which should be constrained to test the theories
with experiments and observations. We construct the kurtosis consistency
relations, calculated based on matter density fluctuations, in which the
information of gravity theories is encoded. We derive two independent
consistency relations that should hold in the framework of the DHOST theories
and argue that such consistency relations would be useful for testing gravity
theories.
