If the concentration of h2 is doubled, will the reaction rate quadruple, double, or remain the same?

The answer depends on the rate law.

1) If the rate law is first order on H2, then:

r = K [H2] / A

Since K is a constant and you are not varying A, then the rate will double if [H2] doubles.

2) If the rate law is second order on H2, then:

r = [H2]^2 / A

Again, since K is a constant and you are no varying A, then the rate will quadruple if [H2] is doubled because:

(2 * [H2]) ^2 = 4 [H2]^2

3) If the rate law is of zero order on H2, then the rate will remain the same, because the rate does not depend of the concentration of H2.