I've made some tests and notice that the browser (at least Firefox and chromium) doesn't send the domain data in the cookie header. Thus the value is not set at the cookie level.
If you really need the domain, you can infer it from the request's reference: request.getResourceRef().getHostDomain()
By doing do, you will get the full name of the request's domain.
If you still need to read the domain from the cookie inside the code of your resources, you can also add a filter in front of your resources that completes the current cookies, and add the domain parameter taken from the request's reference.
// First server response, that requires the browser to record a cookie.