If you want expedited consideration of your writ application, you should file it as soon as possible. You shouldn't exacerbate or (worse) create the emergency by waiting until the last possible day to file. That’s a matter of courtesy and professionalism. It’s also a matter of self-interest: the last thing you want to do as a writ applicant is antagonize the court by dragging your feet on filing and then insist on an immediate ruling by the court.
In some Louisiana courts, it’s also the rule. A few days ago, the Louisiana Fifth Circuit adopted Local Rule 14. The new rule requires a writ application seeking emergency consideration to be “be filed in this court as soon as possible after the lower court ruling but not more than ten days after issuance of the notice of the ruling.” Failure to do so without good cause may be grounds for denial of emergency consideration.”
Since the beginning of this year, the Louisiana Second Circuit has had a similar rule, except that the time to file is 15 days instead of 10 days. See La. 2 Cir. Local Rule 17. Both the Second and Fifth Circuits appear to be following the Louisiana Supreme Court’s example. Under the LASC’s Rule X § 5(a)(2), “An application seeking expedited review of a judgment of the court of appeal as ... shall be filed as soon as possible after the court of appeal’s disposition and in no event more than ten days after the mailing of notice of judgment by the court of appeal.” The penalty for failure to comply “may be grounds for denial of expedited review (with review in regular course if the application is otherwise timely under subsection (a) (1)) and/or imposition of sanctions against the party seeking expedited review.”
But regardless of whether the court you’re filing in has a rule like this, it’s just good practice to file your writ application as soon as possible if you need expedited consideration. The court will appreciate your effort to give them as much time as you can to consider your application. That good will should enhance your chances of winning. At the very least, you’ll avoid sabotaging your own writ application by antagonizing the court.