It is not currently possible to cast via Chromecast on the Live TV feature in the A&E app. You will not see the cast button available in the Live TV section of the Android, iPhone, or iPad apps.
If you are in the U.S., the A&E Live TV stream is available for viewing after signing in with your TV provider on a web browser at play.aetv.com/live and in the A&E apps on Roku, Apple TV, Fire TV and Samsung devices.
While on demand programs are available to A&E users in Canada, Live TV viewing is not currently available outside of the U.S.