You're able to set a cookie so that it works on all subdomains (www, subdomain, etc.). See Basics of Cookies in ASP.NET:
By default, cookies are associated with a specific domain. For example, if your site is www.contoso.com, the cookies you write are sent to the server when users request any page from that site. (Except for cookies with a specific path value, as I explained in the section immediately preceding.)
You can also use the Domain property to create a cookie that can be shared among multiple subdomains. For example, set the domain as follows:
Response.Cookies("domain").Value = DateTime.Now.ToString
Response.Cookies("domain").Expires = DateTime.Now.AddDays(1)
Response.Cookies("domain").Domain = "contoso.com"
The cookie will then be available to the primary domain as well as to sales.contoso.com and support.contoso.com.