No, there are so many problems with bringing a rook out early that no serious opening has that kind of development. Recently I saw a good section in a chess book that described in detail why it was bad idea even if the rook were able to escape to the center live (not captured at a3 or h3 by a bishop): not enough mobility with typical pawn formations, not connected with the other rook, getting in the way of squares to which other pieces need to develop, preventing the ability to castle, etc. If I come across that book again I'll see if I can quote it.
Rook opening?

The main ideas behind a successful opening are to control the centre and devlop your pieces (and a safe King I guess). A strong centre will mean better access to key squares for your pieces in the middlegame.
What you suggest hinders all of these things. Those pawn moves weaken the possible future homes for your King. Getting your Rooks out will allow your opponent to win tempi off of them by playing moves that they want to play anyway (d4/5 & e4/5) through attacks from the Bishops. Your Rooks will then probably end up moving centrally where they will restrict your central pawns and other pieces and again be susceptible to tempo gaining moves.
I recommend that you read this article on the basics of the opening: https://www.chess.com/article/view/the-principles-of-the-opening

The undefeatable rook opening
1-A4 2-D3 3-C3
From there you can knight B1-D2-B3 to protect pawn to A5 if possible so you can bring rook out to A4 if bishop threatens rook to A3

Try the Stafford Gambit. Apparently, you can sometimes use the rook. I’m not exactly someone who should be giving advice, but Eric Rosen has a video on it.

There is only one playable opening in the game of chess where the rook is the first peice to be developed: The Emory Tate Variation of the Alekhines defence: 1.e4 Nf6 2. e5 Nd5 3. c4 Nb6 4. a4 a5 5. Ra3
Hi there, I am "alright" with chess, I like to play the game but I only play occasionaly with friends, and I don't have a standard opening as such. I know that in general you should not employ rook early in the game, however are there any openings that do not weaken your position? And I mean getting the rook out asap so if you played as white your first two moves would be one of the two (h4 followed by Rh3 or a4 followed by Ra3). Are there any advantages for white in such opening? If not, how should black play to get the most out of such opening? It is my very first post on this forum, so if this was asked before, I will gladly look into such post if I can get the link. Thank you. :)