Tex. Civil Practice and Remedies Code § 35.006

(a) If the judgment debtor shows the court that an appeal from the foreign judgment is pending or will be taken, that the time for taking an appeal has not expired, or that a stay of execution has been granted, has been requested, or will be requested, and proves that the judgment debtor has furnished or will furnish the security for the satisfaction of the judgment required by the state in which it was rendered, the court shall stay enforcement of the foreign judgment until the appeal is concluded, the time for appeal expires, or the stay of execution expires or is vacated.

(b) If the judgment debtor shows the court a ground on which enforcement of a judgment of the court of this state would be stayed, the court shall stay enforcement of the foreign judgment for an appropriate period and require the same security for suspending enforcement of the judgment that is required in this state in accordance with Section 52.006.