Has to appoint within 30 days of absence beginning, but appointment lasts until end of term or when Peters decides he can continue, whichever is first.
Texas Local Government Code
Sec. 87.043. Temporary Absence in Office of County Judge in Certain Counties.
....
(b) If a temporary absence exists in the office of county judge, before the 30th day after the date the absence begins, the absent county judge may appoint a resident of the county to fill the office until the next term of that office or until the temporary absence ends, whichever event occurs first. If the absent county judge does not appoint a resident of the county within the 30-day period, the commissioners court shall appoint a resident of the county to fill the office until the next term of that office or until the temporary absence ends, whichever event occurs first.