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.