It's an ego thing. That's why we all get nervous.
Control your ego and you won't get nervous.
Cause the fact that you're nervous about the different piano you're playing on isn't about the piano, or the audience, or the environment, it's about you.
So the way you get around that is by not putting yourself over the music. Instead of thinking, 'Okay this is an unfamiliar piano, I hope this piano doesn't mess me up I don't wanna look bad in front of the audience', you should really be thinking 'Okay I have business to take care of. I gotta bring out the intentions of this composers music to the audience as best I can'
So if you make the music the priority, and not yourself, you trivialize all the external stuff like the different piano, sweaty fingers, big audience etc, and you play better cause you're not as nervous anymore.