You, like me, have just bought an Astro C40 Tournament Controller. I can help you with how to use it on PS5.
Astro C40 for PS4 Controller
The Astro C40 controller is a pretty good pro controller for pro gamers or just really hardcore gamers on PS4 that want to be able to have something similar or close to good as the Xbox Elite Series controller is. Although it was great before the controller, it is now less cool. However, it's still a superior controller streamingliveacademy.com/how-to-use-astro-c40-tournament-controller-on-ps5/ than average.
The issue is that Sony decided to nuke all older controllers in the so called name of progress. Astro, the company, has stated on Reddit through a representative that Sony cannot update the controller. The controller is not working on PlayStation 5 with PlayStation 5 games, but it will work with PlayStation 4 titles .

How To Use Astro C40 Tournament Controller on PS5
Reddit Post
Besides them replying to such a cultured users name I applaud them for replying and letting us know. The link to the post is on the image itself or here. Although I won't be buying a PS5 any time soon, I don't worry. However, I do prefer the feel of the Astro C40 controller to the Sony PS5 controller. The haptic feedback is also very good and can make games more interesting.
My issue with PS5 controller is that it doesn't have paddles and the joysticks are designed in a way that I don't like since I hand down prefer the Xbox Controller joystick layout a lot more. We want to know if you are in agreement by voicing your opinion under SLA and Streaming Live Academy.
Astro C40 allows for Xbox style joysticks on PlayStation natively
There are still hope. As I said at the beginning, I will answer the question "How to Use Astro C40 Tournament Controller on PS5?" You can use the Astro C40 controller on the PS5 while you play PS4 games, but for those PS5 titles you are going to have to use a third party device called the Cronus Zen.
Cronus Zen
This tiny device allows you to use the Astro C40 TR controller natively on your new PS5. I invite you to go read the article of what it is then decide for yourself if purchasing is worth it. Personally, I'll probably do it once I get my PS5 and will share my findings with you via SLA and my YouTube channel.
Sony should realize that not all games will want or use the haptic feedback feature. They seem to be hell bound to making sure that everyone does and honestly I can't blame them for I wouldn't want them to have an Apple 3D Touch situation on their hands and the technology is really cool.

It's not that it's a necessity. Also, it isn’t something you really need for most games. So forcing everyone to use it seems excessive to me. By the time I end up getting a PlayStation 5 they might have already reversed course, but we shall see then. For now all we have is the Cronus Zen and PS4 titles on our PS5 controller life.

How To Use Astro C40 Tournament Controller on PS5
You can only play PS4 (PlayStation 4) games on the PS5 (PlayStation 5) with the Astro C40 TR (Tournament) controller, because Sony has decided to not allow developers of controllers to migrate older controllers that don’t support haptic feedback. You can however use a device like the Cronus Zen to use the Astro C40 TR controller on the PS5 and play PS5 games.