Short Answer: No. An +18V supply will just make the pedal warmer inside. Which may actually make it ever so slightly noisier.
Long Answer: In the real world pedal power supplies are not all made equally. Some are great, and can deliver a nice clean voltage, but other's may be noisy. This noise can translate into noise in the audio signal, not good!
To avoid these problems Empress uses a regulator at the input of the power supply. The regulator takes the power coming in (which can be anywhere between +9V and +18V) and outputs a really clean and constant +8V. This means that the internal electronics of the pedal see a nice clean power supply regardless of what the guitarist powers the pedal with.
We aren't really happy with the headroom of a single +8V supply so in our pedals we double the headroom by using an 'inverting switching regulator', this device takes the +8V and creates a -8V supply. This way we can power the pedal electronics with +8V and -8V which gives 16V of overall headroom. You'll notice this is close to 18V, and typically in a pedal that uses an 18V supply they would regulate down to 15V to 16V.
So, almost magically, we give you the equivalent of 18V of headroom from a 9V supply.