The "states of matter" classification is a useful way to understand matter in broad strokes, but it has limits to its usefulness. In particular, not all phase transitions correspond to changes in the "state of matter" occupied in the broad-strokes classification.
As one simple example, magnetic materials experience a phase transition (known as the "<PERSON> point") when you heat them up: if you start with a ferromagnet with a permanent magnetization and you heat it up, there is a sharp point where it will lose that permanent magnetization. From the point of view of thermodynamics, this phase transition shows the same behaviour as the melting of a solid into a liquid. But would you count "permanent magnet" and "demagnetized magnet" as different "states of matter"? Or are they both just "solid"?
Similarly, there are phase transitions within each "state of matter", particularly within solids. One good example is water ice, which can crystalize into about twenty different crystal structures, depending on the temperature and pressure. 273
And, if you change the temperature and pressure so that you cross the boundary into a different crystal structure, you get a phase transition with the same thermodynamics again.
To make things worse, even the clear separation lines in our paradigm can get blurred. In normal conditions, liquids and gases are distinct and clearly recognizable, and you have a thermodynamic phase transition (evaporation / condensation) when you go from one to the other. But, if the temperature is high enough, you can get something called a supercritical fluid which has properties of both, and which can connect smoothly to both a liquid and a gas when you cool it down, depending on the pressure, without ever passing through a phase transition. So, should liquids and gases be considered as just one single "state of matter"?
OK, so, that said $-$ what about topological phases? This is a hard question to answer, because each topological phase is (in some ways) specific to the individual system where it appears, while also (in other ways) very easily identifiable across a range of different systems. There is a rigorous classification of the different topological invariants that characterize topological phases, colourfully referred to as the periodic table of topological invariants, and this gives a good sense of what is possible in this area: probably around one or two dozen, or several hundred, depending on whether you count each phase as corresponding to one individual invariant or to a full specification of several. Again, this isn't an area where any hard-and-fast statements can really pull any weight, and there's a heck of a lot more physics to discover and understand before we can be sure that we've found everything, but that "periodic table" is a good picture of our current understanding. 28