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In this paper we revisit the topic of Hawking radiation as tunneling. We showthat the imaginary part of the action of the tunneling particle should bereconstructed in a covariant way, as a line integral along the classicalforbidden trajectory of tunneling particles. As the quantum tunnelingphenomenon, the probability of tunneling is related to the imaginary part ofthe action for the classical forbidden trajectory. We do the calculations formassless and massive particles, in Schwarzschild coordinate and Painlev\'ecoordinate. The construction of particle action is invariant under coordinatetransformations, so this method of calculation black hole tunneling does nothave the so called "factor 2 problem". As an application, we find that thetemperature of Hawking temperature of apparent horizon in a FRW universe is$T=\frac{\kappa}{2\pi}$. Based on this result, we briefly discuss the unifiedfirst law of apparent horizon in FRW universe.

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