  We investigate the potential of cosmological observations, such as galaxy
surveys, for constraining degenerate higher-order scalar-tensor (DHOST)
theories, focusing in particular on the linear growth of the matter density
fluctuations. We develop a formalism to describe the evolution of the matter
density fluctuations during the matter dominated era and in the early stage of
the dark energy dominated era in DHOST theories, and give an approximate
expression for the gravitational growth index in terms of several parameters
characterizing the theory and the background solution under consideration. By
employing the current observational constraints on the growth index, we obtain
a new constraint on a parameter space of DHOST theories. Combining our result
with other constraints obtained from the Newtonian stellar structure, we show
that the degeneracy between the effective parameters of DHOST theories can be
broken without using the Hulse-Taylor pulsar constraint.
