  With the recent progresses on the Type II supernovae, we attempt to
investigate whether there does exist new physics beyond the standard
cosmological paradigm, i.e., the cosmological constant $\Lambda$ plus cold dark
matter ($\Lambda$CDM). Constraining four alternative cosmological models with a
data combination of currently available Type II supernovae calibrated by the
standard color method, Type Ia supernovae, baryon acoustic oscillations, cosmic
microwave background and cosmic chronometers, at the $1\sigma$ confidence
level, we find that: (i) a spatially flat universe is supported for the
non-flat $\Lambda$CDM model; (ii) the constrained equation of state of dark
energy $\omega$ is consistent with the $\Lambda$CDM hypothesis for the
$\omega$CDM model where $\omega$ is a free parameter; (iii) for the decaying
vacuum model, there is no evidence of interaction between dark matter and dark
energy in the dark sector of the universe; (iv) there is also no hint of
dynamical dark energy for the dark energy density-parametrization scenario. It
is very obvious that a larger Type II supernovae sample is required, if we
expect to draw definitive conclusions about the formation and evolution of the
universe.
