Invariance of the Hamilton-Jacobi tunneling method for black holes and FRW model

In this paper we revisit the topic of Hawking radiation as tunneling. We show that the imaginary part of the action of the tunneling particle should be reconstructed in a covariant way, as a line integral along the classical forbidden trajectory of tunneling particles. As the quantum tunneling phenomenon, the probability of tunneling is related to the imaginary part of the action for the classical forbidden trajectory. We do the calculations for massless and massive particles, in Schwarzschild coordinate and Painlev\'e coordinate. The construction of particle action is invariant under coordinate transformations, so this method of calculation black hole tunneling does not have the so called "factor 2 problem". As an application, we find that the temperature of Hawking temperature of apparent horizon in a FRW universe is $T=\frac{\kappa}{2\pi}$. Based on this result, we briefly discuss the unified first law of apparent horizon in FRW universe.