  In this work, we study the renormalization group invariance (RGI) of the
recently proposed covariant power counting (PC) scheme in the case of
nucleon-nucleon scattering [Chin.Phys. C42 (2018) 014103] at leading order
(LO). We show that unlike the LO Weinberg case, RGI is satisfied in the
$^3P_{0}$ channel, because a term of $pp'$ appears naturally in the covariant
PC scheme at LO. Another interesting feature is that the $^1S_{0}$ and
$^3P_{1}$ channels are correlated. Fixing the two relevant low energy constants
by fitting to the $^1S_{0}$ phase shifts at $T_\mathrm{lab.}=10$ and $25$ MeV
with a cutoff $\Lambda$ of $400-650$ MeV, the $^3P_{1}$ phase shifts can be
described relatively well. In the limit of $\Lambda\rightarrow \infty$, the
$^1S_0$ channel becomes cutoff independent, while RGI is lost in the $^3P_{1}$
channel, consistent with the Wigner bound and the previous observation that the
$^{3}P_1$ channel better be treated perturbatively. As for the $^1P_{1}$ and
$^3S_{1}$-$^3D_{1}$ channels, RGI is satisfied, similar to the Weinberg
approach.
