How can I prevent the cheese on my homemade pizza from hardening too soon?
The easiest solution is to use different cheeses.

Most commercial pizzerias, like Domino's or Pizza Hut do not use expensive cheeses like Parmesean or fresh mozzerella... they use crappy cheese designed to be stretchy and to stay that way when warm instead of hot. In general, they use part-skim mozzarella, which is often sold pre-shredded and in hard blocks like you'd find cheddars. This is addressed in another answer, here. 

Hard cheeses like Parmesan are unlikely to melt at all and, when they do, they will firm up quickly as they lose heat. You should rarely expect a hard cheese to melt well. Most of them don't. They also lose a lot of oils when they melt, causing unsightly separation. They're great for a burst of salty cheese flavor but they won't behave like Domino's cheese. This is why it's common to top a completed pizza with shredded Parmesan but you wouldn't want to use it as your primary base cheese for the unbaked pizza.

Fresh cheeses like ricotta and mozzarella have the opposite problem. Their high moisture content will often make pizza unappetizingly soggy.

In contrast, the part skim mozzarella used in pizzerias is specifically chosen because it has low water content.

Check out the two questions linked here for a ton more info.

You should consider doing some research into which cheeses melt well as opposed that separate into an gross mess. Note, this list, like many others includes Asiago but it's important to notice that the list refers specifically to fresh Asiago, not the Parmesan-like aged sibling.