  We perform a numerical study to recognize the difference between various
massive potentials in the dRGT massive gravity on the holographic
thermalization in the AdS and AdS Gauss-Bonnet gravities. The massive potential
in $4+1$ dimensions includes three symmetric polynomial terms which we denote
them as $a_1$, $a_2$ and $a_3$ terms. We observe, in the case of time evolution
of entanglement entropy that there is a critical size of the entangling surface
on the boundary below which both signs of $a_1$ and above the critical size
$a_3$ are able to reduce the thermal value of entanglement entropy. Our
numerical computations show the more positive $a_i$'s are, the faster system
reaches to its thermal value. The order of saturation time of positive
potentials when supplemented to AdS or AdS-GB backgrounds is as
$t_{sat}(a_{1})>t_{sat}(a_{2})>t_{sat}(a_{3})$. We also explore these effects
on the time evolution of the holographic mutual information.
