Can shape of the universe change? Is it possible for a closed [spherical geometry] Universe to evolve to become an open [hyperbolic geometry] Universe?
 A: You may have heard that the universe is considered a closed system. This is closely related to how we define the universe. We say that a system is closed if we can't have any transfer of mass between it and any other system. This certainly holds for the universe since any other potential system to transfer anything to would by definition also be part of the universe. Hence the universe is considered closed independently of the evolution of it.
A: This is really an interesting question. Can the universe that is spherical change its topology into a flat open sheet, or a variant on that as a torus that is also flat with a repeating structure. In other words if you travel on a torus far enough out you come back to the same place just as in some video games exiting the top or sides of the screen results in the avatar emerging at the bottom or other side. Can this happen? I will first argue that it can't, but the point to how this might have loopholes or flaws. 
A flat space is simply connected. A sphere $\mathbb S^n$ of dimension $n$ is locally simply connected. The difference between the two is being their homologies. The homology of a sphere is $H_n(\mathbb S^n,~\mathbb Z)~=$ $H_n(\mathbb S^n,~\mathbb Z)~=~\mathbb Z$, and $H_i(\mathbb S^n,~\mathbb Z)~=~0$, $0~<~i~<~n$, For the flat space $\mathbb R^n$ was have a similar topology but with $H_n(S^n,~\mathbb Z)~=~0$. For a space embedded in spacetime $n~=~3$. I will use the same argument for the case of a torus $\mathbb T^n$ where the topology of the $\mathbb R^n$ is changed with $H_i(\mathbb T^n,~\mathbb Z)~=$ $\mathbb Z\times\mathbb Z$ for $i$ odd. 
The case of the torus is similar to a wormhole, which has the topology $\mathbb S^1\times\mathbb R^2$, and in both of these cases on the level of geometry transformations, Lorentz boosts of wormhole opening etc, can turn this system into a spacetime with closed timelike curves. What this then means is that a quantum state can be sent on a closed timelike curve and returned to the experimenter prior to its being sent. This is a way that quantum states can be duplicated. I will then appeal to the no-cloning theorem of quantum mechanics to say this is forbidden. Consequently a torus type of topology is ruled out, at least ruled out for causal curves that can return to the same point in space.
Can a sphere $\mathbb S^n$ be converted into $\mathbb R^n$? Maybe, and the reason this might be possible is that can happen. I worked up a model with a quantum critical point where an initially spherical space in preinflationary cosmology inflates into a flat space. This means there are departures from local and global states. In other words local states on the sphere obey the same physics as those on the flat Euclidean space. This might then be a way that quantum states are not completely duplicated on a closed curve. There is in other words no time machine or closed timelike curve problem. 
This is not an entirely well understood problem. It hinges upon the relationship between local and global principles of physics and cosmology vs physics. It is curiously similar to symmetry protected topological states, where quantum states on a boundary can have properties that appear to violate conservation laws, such as chirality of electrons, but where topologically or in the bulk (think of a Dirac monopole or string) this violation is not there. 
