Perhaps unsurprisingly, given their higher price than the standard AirPods, it'll cost more for you to repair or replace the new AirPods Pro through Apple Support. Out of warranty, the price this time around is $89 per earbud, and it costs the same to have the wireless charging case repaired or replaced. If you lose an AirPods Pro earbud, Apple charges $89 for a new one or $99 to replace a misplaced case.
For comparison, it costs $69 to replace or repair an AirPod, $59 for the regular case and $69 for the wireless charging case. AirPods Pro cost $249, while AirPods are priced at $159 or $199, depending on whether you opt for the wireless charging case.
However, the replacement and repair costs drop quite a bit if you have AppleCare+ for Headphones (which costs $29 for two years). Under that plan, Apple charges $29 for each repair or replacement of all its earbuds and cases.
Update 10/29/2019 1:15PM ET: Noting that AppleCare+ for Headphones lasts from two years.