Can I add multiple default gateways for BIG IP ASM 3600 v11.1.0?

You can use default wildcard virtual server (vs for net 0.0.0.0/0 type fwd l4 with a pool pointing to alternate gateway) operating on a specific vlan. Thats good for hosts using big-ip as default gateway.

If the need is have multiple external gateways, create a vlan and virtual servers for each and let auto last hop feature deal with it... connection will obey it instead of default route. Better than ... go for GTM and have fun !