>>1
Last I checked, the universe is very nearly flat. If there is any deviation at all from flatness, the deviation will grow exponentially with time. After ~ 10 billion years, that deviation should have grown to be huge, unless the initial condition was a ridiculously small deviation.
As for evidence, type Ia supernovas, and maybe cmb place weak bounds on the flatness (keep in mind that this weak bound is now, and not when the universe began; see above note). There might exist better evidence of flatness.
Even if the universe were asymptotically AdS, I remember there exists a slicing of AdS that is spatially flat. I think the "flatness" refers to the spatial slices.