### [REQUIRED] Environment info

**firebase-tools:** 12.4.7

**Platform:** macOS

### [REQUIRED] Test case

When using Firebase Auth with a Next.js App Router (now the recommended router over Page Router), can't access the name of the initialized Firebase app as recommended in [the docs](https://firebase.google.com/docs/hosting/frameworks/nextjs#optional_integrate_with_firebase_authentication).

### [REQUIRED] Steps to reproduce

1. `npx create-next-app@latest`
1. when asked _Would you like to use App Router? (recommended)_, answer "Yes"
1. `firebase init hosting`
2. Access `searchParams` either in a server component or client component:

server component:

```tsx
import { getApp } from "firebase/app";

export default async function Home({
searchParams,
}: {
searchParams: { [key: string]: string | string[] | undefined };
}) {
return (

searchParams: {JSON.stringify(searchParams ?? "not found", null, 2)}

);
}
```

or client component:

```tsx
"use client";

import { getApp } from "firebase/app";
import { useSearchParams } from "next/navigation";

export default function SignInButton() {
const firebaseApp = useSearchParams().get("__firebaseAppName");
return (
<>

Client component

authenticated Firebase App name (from{" "}
useSearchParams().get("__firebaseAppName"))

__firebaseAppName: {JSON.stringify(firebaseApp)}

);
}
```

### [REQUIRED] Expected behavior

`__firebaseAppName` is available, as documented: https://firebase.google.com/docs/hosting/frameworks/nextjs#optional_integrate_with_firebase_authentication

### [REQUIRED] Actual behavior

`__firebaseAppName` isn't available on the server or client

Start by reproducing the issue with `npx create-next-app@latest`, choosing App Router, then running `firebase init hosting` and checking `__firebaseAppName` through the documented server and client examples. Trace the Firebase Hosting Next.js integration used by that command; done means the documented parameter is available in both contexts or the documentation accurately reflects the supported behavior.

Written by the indexing model from the issue text.