"Price" is the best choice in your example, but "cost" would not be incorrect.
Both words play such big parts in our lives, they have many specialized meanings, some technical, some figurative. You really need to add some context.
Giving up drinking was too high a price to pay to save his marriage.
The cost of his drinking to his marriage was unbearable.
The price of gas is cheaper in the next town, but the cost of going there to get it is too high.
"Price" is usually more specific.
"Cost" is usually something that has to be "figured out."