Apple has been extremely helpful and a very good partner in root causing the source of this problem.
We are pleased to announce that we have made available a temporary workaround (CSCuh73010) for the AnyConnect Mac OS X 10.9 connection instability / compatibility issue (Apple Bug Report ID 15261749) via an updated version of AnyConnect (3.1.04074). There are caveats associated with this temporary workaround which are noted in the release notes. Due to replication delays, the release notes may not be updated until Monday.
http://software.cisco.com/download/release.html?mdfid=283000185&flowid&softwareid=282364313&release=3.1.04072
(It is located in the 4072 directory.)
Release Notes:
http://www.cisco.com/en/US/products/ps10884/prod_release_notes_list.html
Note: A SMARTnet-enabled login is required to download AnyConnect Software from Cisco.com.
Here's the release note excerpt: This update is a temporary workaround to resolve the problem of frequent disconnects of the AnyConnect VPN on systems running Mac OS X 10.9. Apple is aware of this issue and you can reference Apple Bug Report ID 15261749 if you want to open your own case with them. This patch is a workaround and we are not able to guarantee that it will work as reliably as a fix provided by a future OS patch. A future OS fix will not require you to update AnyConnect 3.1.0472 to operate reliably. Once Apple provides a fix in OS X 10.9, we may choose to retract this workaround, and unpatched versions of AnyConnect 3.1.04072 will also work reliably.
Best Regards,
Pete Davis
Product Owner, Cisco AnyConnect
ac-mobile-feedback@cisco.com
@AnyConnect
http://facebook.com/anyconnect