For the early universe(R<1) there is little difference among the kinematic behaviors of a flat, a closed, and open universe because the early universe was essentially flat.
Indeed somewhat confusing, because if the early universe were essentially flat, how can it develop in a closed, flat, or open universe? Won't it stay flat forever?
f you imagine the universe to be a balloon (reducing the number of space dimensions to two), it is said that for every model (closed, flat , or open) the developing balloons are identical in the early universe (around $T=1$). I'm not so sure what is meant by essentially flat means that the balloons locally don't differ from a flat spacetime (though their overall form is that of a balloon). This is also the case for the three types of universe in the distant future.
So while the balloons don't differ much from each other now, they will differ from each other in the future.
If you take accelerated expansion seriously you can nothing but conclude that we are living in an open universe (the balloon will not only expand forever, but it will do so in an accelerated way (though it remains to be seen if the accelerated expansion will continue forever). Before the accelerated expansion started the universe looked locally flat.