  We investigate cosmological constraints on the original relaxion scenario
proposed by Graham, Kaplan and Rajendran. We first discuss the appropriate sign
choice of the terms in the scalar potential, when the QCD axion is the relaxion
with a relaxion-inflaton coupling proposed in the original paper. We next
derive the cosmologically consistent ranges of the mass and a coupling of the
relaxion for both the QCD relaxion and non-QCD relaxion. The mass range is
obtained by $10^{-5}$ eV $\ll m_{\phi} \lesssim 10^4$ eV. We also find that a
strong correlation between the Hubble parameter at the relaxion stabilization
and the scale $\Lambda$ of non-QCD strong dynamics, which generates the
non-perturbative relaxion cosine potential. For a higher relaxion mass, a large
scale $\Lambda$ becomes available. However, for its lower mass, $\Lambda$
should be small and constructing such a particle physics model is challenging.
