  The beyond-Horndeski gravity has recently been reformulated in the dark
energy paradigm - which has been dubbed, Unified Dark Energy (UDE). The
evolution equations for the given UDE appear to correspond to a
non-conservative dark energy scenario, in which the total energy-momentum
tensor is not conserved. We investigate both the background cosmology and, the
large-scale imprint of the UDE by probing the angular power spectrum of galaxy
number counts, on ultra-large scales; taking care to include the full
relativistic corrections in the observed overdensity. The background evolution
shows that only an effective mass smaller than the Planck mass is needed in the
early universe in order for predictions in the given theory to match current
observational constraints. We found that the effective mass-evolution-rate
parameter, which drives the evolution of the UDE, acts to enhance the observed
power spectrum and, hence, relativistic effects (on ultra-large scales) by
enlarging the UDE sound horizon. Conversely, both the (beyond) Horndeski
parameter and the kineticity act to diminish the observed power spectrum, by
decreasing the UDE sound horizon. Our results show that, in a universe with
UDE, a multi-tracer analysis will be needed to detect the relativistic effects
in the large-scale structure. In the light of a multi-tracer analysis, the
various relativistic effects hold the potential to distinguish different
gravity models. Moreover, while the Doppler effect will remain significant at
all epochs and, thus can not be ignored, the integrated Sachs-Wolfe, the
time-delay and the potential (difference) effects, respectively, will only
become significant at epochs near z=3 and beyond, and may be neglected at late
epochs. In the same vein, the Doppler effect alone can serve as an effective
cosmological probe for the large-scale structure or gravity models, in the
angular power spectrum - at all z.
