After the actual acceleration there wouldn't be any difference to how life on such planets works. What's difficult for me to estimate is the likelihood of surviving the slingshot itself however. Looking at the closest known orbit around Sagittarius A* [1] you get an orbital acceleration of this star system of around 0.2G. I'm not entirely sure about the math, but I'd imagine the slinghot would have to keep an orbital acceleration of significantly less than the planet's own surface gravity, otherwise it would probably strip its atmosphere.

[1] https://en.wikipedia.org/wiki/S2_%28star%29