Unless it's a tiny class, probably not (see this question for possible pitfalls with large sessions). In short, sessions are not designed to be a caching mechanism, and they don't perform too well when you make them into one.
Note that if you are using the default session handler, your sessions are stored on the hard drive - not very fast when you get many concurrent requests. Also (test and measure), serialization/deserialization may be slower than the normal methods of object creation - note that you'd probably be deserializing twice: from session to string, then string into object of that class.
If you want to go the serialization/deserialization route, try e.g. Memcached instead.