What happen if the rolled dice is doubled and another gate is opened by the new mythos card?

I mean, if so, Should I draw another mythos card again by the Zhar's specialty?

I've always played that when a new gate opens as a result of drawing a mythos card after rolling doubles, the dice get rolled again. If a gate opens, then roll the dice again. Etc, etc, etc.

While slim, the possibility does exist that you could roll doubles often enough to wake up Zhar on the first turn. Not sure it's ever happened, and if it ever does it would make quite a story to tell.

Yup, his power says to roll when a new gate opens, if a gate is opened by the resulting mythos card then a new gate has opened, so roll. It's actually far more likely that Zhar will run a bunch of monsters on the Innsmouth board into vortexes driving up the DOR track, this is made more likely if you team him up with Dagon and Hydra.