Johnny, what you see are packets that are part of the outgoing connections from you computer to Akamai. Most probably these are packets belonging to connections that have just been terminated. Avast FW shouldn't in fact be reporting these, those might be retransmited packets or RST packets that would normally be part of the traffic (or happen after the traffic was terminated). I guess we are incorrectly reporting these here (and possibly also incorrectly block them) when in fact the FW should for some short period of time (just after the connection has been terminated) allow these packets to go through - since they are regular part of the TCP traffic, albeit ending one. In our internal builds, this behavior has already been improved.
Your fear about Akamai using your PC as a backup host is completely false. They don't do this. Akamai is well respected content delivery network (one of the largest in the world), used by large companies (such as Microsoft - as mentioned before) and secondly, these are really outgoing connections, so they were initiated by your computer and not the other way around.
Blocking Akamai, as suggested by polunus, unless you have strong reasons to do so, might disrupt all kinds of things during your regular PC usage (starting from Windows update, going via various software updates - adobe reader, flash, etc. and ending with various web sites). Even Avast, I guess, might host some (static) web content on Akamai servers - they have many servers all around the world, so that you can always connect to the one closest and fastest to you - that's what this service is about.
To sum up: most probably these are regular packets that should not be reported in this log. We are trying to figure what these cases are and fix them for the upcoming builds. Sorry.
Lukas.