The CDS manages one cookie. This cookie is called
__saml_idp and its use and format is described in section 4.3.1 Common Domain Cookie of the SAML2 profile specification. The function of the cookie is to record previously selected IdPs. The cookie may be session or permanent depending on the GUI (which may itself devolve responsibility to the user).
It should be noted that the Java container hosting the CDS may also use one or more cookies (usually session) to record session state. Refer to the container's documentation for details.