Changing the mass of the pendulum bob has two opposing effects.
A pendulum of double the mass experiences double the gravitational force (and double the pull from the cord) - but it has twice the inertia (i.e. it requires double the force to start and stop it moving in the same way).
These two effects cancel out - leaving the pendulum's behaviour almost unchanged.
Almost, because doubling the pendulum's weight stretches the cord a little - making the pendulum longer. And length does affect the period of its swing.