  We present a first-principles implementation of the stochastic path-integral
approach proposed by Liu et al. [H. Liu, Y. Yuan, D. Liu, X.-Z. Li, and J. Shi,
Phys. Rev. Research 2, 013340 (2020)] for estimating the superconducting
transition temperature ($T_c$) of a liquid. The implementation is based on the
all-electron projector-augmented-wave (PAW) method. We generalize Liu et al.'s
formalism to accommodate the pseudo-description of electron states in the PAW
method. A formula for constructing the overlap operator of the PAW method is
proposed to eliminate errors due to the incompleteness of a pseudo-basis set.
We apply the implementation to estimate $T_c$'s of metallic hydrogen liquids.
It confirms Liu et al.'s prediction that metallic hydrogen can form a
superconducting liquid.
