Is there any problem to use bgp private AS when multihoming to different ISPs and advertising the IP addresses assigned by one of the ISPs. On RFC it says that Private AS should be used only when singlehomed.
Can someone give me any explanation about the disadvantages of using Private AS in this case?
Well, the private AS # will not be propagated between the different ISPs. As BGPs loop detection algorithm is to watch for its own AS# in announcements, you run the risk of looping, using transit to try route between your border routers (remember, eBGP has admin distance of 20, iBGP has admin of 200), etc. This is not a good thing! (you could mitigate much of this with carefully applied route-maps that block receiving your own prefixes).
People receiving your announcements will also see inconsitent source ASs for your announcments and so you run the risk of some people not installing them.