This reaction must be 2nd order. The half-life expression for a 2nd order reaction is:
A --------->B
t1/2 = 1 / [A] x k
t1/2 = 1 / [2A] x k = t1/2 = (1/2) x 1/ [A] x k
Here we see that, if [A] is doubled, the half-life is halved.
The reaction is 2nd order.