Anthony Appiah has a very simple to understand but at the same time well-founded sense of cultural plurality and tolerance, resumed in the word Cosmopolitanism, a word that has a specific meaning in philology or in the daily use of words/ language, and of which Appiah appropriate to give it another specific meaning in his own terms, related with the usual meaning but with a note of social and philosophical complexity.
To build his notion of Cosmopolitanism, Appiah departs from post-modern pragmatism, to ask the public and himself how could we invest the sense of citizenship with an actual meaning in the globalized World of our days (and the question itself reveals a sense of pragmatism, since he is not asking about the truth behind the concept of citizenship or its foundations, but about what we can do with a tool —citizenship— like this). We have the task of seeing ourselves like citizens not only within the societies in which we take directly part (our neighbourhood, our country, even our continent), but within the whole World, since almost virtually every country and every culture in the World is interacting in any way with the others (well, we may exclude some populations on the Amazonia, for example, but actually the are the exception to the rule). We cannot simply reject or violently react against such cultural different, since there is something which we have to deal with everyday, we like it or not, and active confrontation only can take us to chaos.
The answer for Appiah is a sort of cultural relativism: we have to say to ourselves that our culture/religion/education is not necessarily the best possible or, in platonic terms, the truth. Our cultural background is not the measure for all things. 1) If everybody thought the same it only could drive to perpetual confrontation and 2) It is not necessary that everybody be the same for the World to work right (again: pragmatism).
Appiah says: “Cosmopolitans think human variety matters because people are entitled to the options they need to shape their lives in partnership with others.” 
Cosmopolitanism is tolerance but not free tolerance, but a tolerance founded on rationalism and pragmatism. Cosmopolitan respect the acts of the cultural other, but not necessarily celebrates it, so Cosmopolitanism as defined by Appiah shouldn't be taken as total cultural relativism or multiculturalism (to think everything is equal good or right). The tolerance of the Cosmopolitan is an acceptation of the other, not necessarily an approbation or the though that the other is intrinsically good. It is, again, pure pragmatism (and a Cartesian attitude regarding to the validity of our own culture).
You could agree or not with Appiah, but we should admit that at least his philosophy is not difficult to understand at all, and that's so much in an epoch of philosophers without clear systems...
Rorty says: “Rationality is a product of participation in a community; rational behaviour is just adaptative behaviour, in similar circumstances of the other members of some relevant community. Irrationality in both physics and ethics is a matter of behaviour that leads one to abandon or be stripped of membership in some such community.” 
Well, parts like “rational behaviour is just adaptative” put Rorty as a clear precedent of the propositions by Appiah, especially with the introduction of the concept of post-modern pragmatism and its social concerns. Rorty introduces relativism against truth, but not a sort of relativism that makes everything the same good or valid, but the relativism that leads to the predominance of the useful against the truth. Epistemology is no more on the foreground, but social utility.
I actually do not know what Rorty would think about Appiah conclusions or statements, but anyway I am pretty sure that such thinking (Appiah's) is directly implied or deduced from the philosophical proposals by Rorty when facing multicultural interaction in the World of today.
On the other hand, Appiah is against cultural etiquettes, following him, the same culture can be pluralistic at the same time that maintaining its roots. That is actually the History of the cultural evolution of the human being. A closed or limited vision of a culture is against Cosmopolitanism in the terms by him defined.
In that sense, we could mention Judith Butler, since such conception by Appiah is very similar to what Butler proposes for gender. Butler is nor necessarily a pragmatist neither a relativist (at least not the Rorty style) but we can see in her way to conceive gender and gender within society close similarities to the conception of cultural and cultural plurality by Appiah. Cultural etiquettes, that in fact are against cultural plurality and evolution, are equivalent to the gender/ sexual etiquettes that act as social constraints for a natural sexual development in the analyses by Butler, since gender/sex is to Butler organic and developing, the same as culture for Appiah. If gender is the result of improvisation in a scene of constraint (Butler words), then we could see such improvisation as the equivalent to gender for Cosmopolitanism as defined to Appiah regarding the different cultures of the World.
- Appiah, Anthony “Cosmopolitan Contamination” from Cosmopolitanism: Ethics in a World of Strangers. 2006.
- Rorty, Richard. “Postmodern Bourgeois
Liberalism”. The Journal of Philosophy
Vol. 80, No. 10, Part 1: Eightieth Annual Meeting of the American Philosophical Association, Eastern Division. 1983
See also: Butler, Judith. “Undoing Gender”. 2004.