  We study the phenomenology of the beyond Horndeski class of scalar-tensor
theories of gravity, which on cosmological scales can be characterised in terms
of one extra function of time, $\alpha_{\rm H}$, as well as the usual four
Horndeski set of free functions. We show that $\alpha_{\rm H}$ can be directly
related to the the damping of the matter power spectrum on both large and small
scales. We also find that the temperature power spectrum of the cosmic
microwave background (CMB) is enhanced at low multipoles and the lensing
potential is decreased, as a function of $\alpha_{\rm H}$. We find constraints
on $\alpha_{\rm H}$ of order ${\cal O}(1)$ using measurements of the
temperature and polarisation of the CMB, as well as the lensing potential
derived from it, combined with large scale structure data. We find that
redshift space distortion measurements can play a significant role in
constraining these theories. Finally, we comment on the recent constraints from
the observation of an electromagnetic counterpart to a gravitational wave
signal; we find that these constraints reduce the number of free parameters of
the model but do not significantly change the constraints on the remaining
parameters.
