I have a character benefitting from cover (4) and another character shooting him with a weapon that has Pen 4.

Do I apply the penetration once for the cover /bypassing it completely/ and then again on the character? Or do I apply the penetration only once?

Penetration is applied only the once. Assuming the body location struck is one protected by the cover. Otherwise, the cover doesn't help.

This is why taking cover is highly useful.

Can you point me to where can I look it up in the book or is it just the way your group uses it? I've spent a lot of time searching and found no clarification on the matter...

Ive never found clarification, but look at it like this.

You have....flak armor. 4AP. You hide behind rockrete, 16,AP. That guy shooting you has a gun with pen 10 because its a meltas or something horrifying like that. He rolls, does 14 wounds. The melta blast has to go through the rockrete and the flak together, so it has to break 20 AP before it touches you.

Yeah, I saw it the same way - Armor and Cover are added together, then the weapon damage/pen is applied.

Same way a suppression shield adds to armor. Same way all armor stacking works. Sum it all up.