COVID-19 Cost Share Adjustment Relief Act of 2020 This bill provides for a 100% federal cost share of assistance provided under the emergency declaration issued by the President on March 13, 2020, relating to COVID-19 (i.e., coronavirus disease 2019) and any subsequent major disaster declaration that supersedes it. The President may provide assistance for activities, costs, and purchases of states or local governments or the owners or operators of eligible private nonprofits, including, among other things backfill costs for first responders and other essential employees who are ill or quarantined, increased operating costs for essential government services due to such emergency, medical equipment and public health costs, and costs associated with maintaining alternate care facilities or related facilities currently inactive but related to future needs tied to the ongoing pandemic event.