The most obvious reason is that companies (public companies, anyway) should care for the very reason that investors care. And the corollary to that is that companies should care not only because investors do, but because investors care enough that this secondary market in sustainability info is springing up.
Firms that provide investors with financial information think they can make money by supplying investors with this information (and just so we're clear, these are large institutional investors, not mom and pops).
So companies also need to care because they need to have information on their sustainability program easily accessible to these information suppliers. If they don't, they will sooner or later stick out like a sore thumb among companies who do.
But there are other reasons, self-interested reasons, why companies should care:
So... what is your company doing about this latest litmus test for corporate health?