There's no such rule. We cannot be sure that the transaction cost will go up with the value. You can see that in Jul 2017 the fees skyrocketed to very high levels but the price was just 3000 dollars atthat time and the number of users was much lower. So we had lower price, lower number of users, but higher fees!

Transaction cost usually goes up when the number of transactions goes up and it's true that the more Bitcoin is worth the more people are willing to pay to get a fast transaction but we don't know if the network doesn't get faster with the introduction of secondary layers like the LN.