  We study the cosmology of an extended version of Horndeski theories with
second-order equations of motion on the flat
Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) background. In addition to a
dark energy field $\chi$ associated with the gravitational sector, we take into
account multiple scalar fields $\phi_I$ ($I=1,2\cdots,N-1$) characterized by
the Lagrangians $P^{(I)}(X_I)$ with
$X_I=\partial_{\mu}\phi_I\partial^{\mu}\phi_I$. These additional scalar fields
can model the perfect fluids of radiation and non-relativistic matter. We
derive propagation speeds of scalar and tensor perturbations as well as
conditions for the absence of ghosts. The theories beyond Horndeski induce
non-trivial modifications to all the propagation speeds of $N$ scalar fields,
but the modifications to those for the matter fields $\phi_I$ are generally
suppressed relative to that for the dark energy field $\chi$. We apply our
results to the covariantized Galileon with an Einstein-Hilbert term in which
partial derivatives of the Minkowski Galileon are replaced by covariant
derivatives. Unlike the covariant Galileon with second-order equations of
motion in general space-time, the scalar propagation speed square $c_{s1}^2$
associated with the field $\chi$ becomes negative during the matter era for
late-time tracking solutions, so the two Galileon theories can be clearly
distinguished at the level of linear cosmological perturbations.
