…then I should be able to play the pet rescue as soon as I open the app.
Instead, if I open GoW after getting the notification, I still have to play a PvP/Explore match for the game to realize there’s a Pet Rescue event. This is a clear case of the left hand not knowing what the right is doing. It’s bothersome.
You can just go and click on “Game” icon manu?
For me, the pet won’t show under the game tab unless I’ve already fought a battle, as @Slypenslyde relates. After that, no problem.
If I click on the “Game” menu, it shows me the last pet rescue that was available. I can’t do squat until I play a match.
While this is not properly addressed have you tried to use a gold key? I feel like i keep repeating this too much, but it’s usually the quickest way to “wake up” the game servers and get stuff to show properly. I hope it works until the situation is fixed.
I know all the workarounds and gyrations.
The excuse used to be, “Well, you have to do a thing that causes a handshake with the server, then your client can know.” Just like tribute.
But now the game tells me there is a pet rescue available. Then I open it up and… it has no clue there’s a pet rescue.
I know and until it’s addressed, in 3.7 Patch probably, i’m just suggesting something to maybe minimize your trouble, not trying to be dismissive over a feature problem.
That’s weird. I have all the notifications turned off, just going to collect tribute every hours. And if there is a pet rescue available, pop-up will happened right as I get to the map, so I can go play it right after collecting tributes.
I’ll pass this along. 
I would like a notification when i log in
“Welcome back ricky, while you were offline you missed 5 rescue pet!! You suck!!!”