Does the pity timer build up across different summoning banners?
Solution 1:
Each banner tracks this separately. You can see your increased odds in each banner by going to "appearance rates", so you can easily track which banners you have increased odds on.
This increments every time you go a round of five summons without summoning a 5* hero. The appearance rate is increased 0.5% for each set of five times.
You can see more information in each banner by going to Details -> More Key Information about Summoning
Additionally, if you somehow manage to summon 120 times without getting a 5*, the rate is increased to 100%.