Welcome to the wonderful world of S0. Basically what you will want to do is either enable S3 support, OR make S0 network disconnected. There are posts on this forum talking about these points.
In essence, S0 is a low power idle state that allow some work to be done. S3 is suspend to RAM with the CPU effectively off and no work happening. S0 does permit a suspend then hibernate scheme based on drained battery percentage. However, you will never see this with the laptop plugged into a power source. S3 will suspend (using less power while doing it, compared to S0) but will suspend right until the battery is dead and the whole system runs out of power.
I just use S0 network disconnected, and if I know I am going to be a while, I just hibernate to keep things cool. YMMV.