It’s officially an “application for pretrial writ of habeas corpus”; denials of such applications can be appealed before trial, and the appeals are to be “heard at the earliest practicable time.” See, e.g., Tex. R. App. P. 31; Ex parte Taylor (Tex. Ct. Crim. App. 2002).

You can read the application here — I’ll be reading it myself soon, and hope to post more about it.