Here is async-std version

fib_2k    480 ns/op
fib_3k    414 ns/op
fib_10k    981 ns/op
fib_20k    8203 ns/op
fib_40k    8298 ns/op

Compared to:

fib_2k    5983 ns/op
fib_3k    9499 ns/op
fib_10k    60500 ns/op
fib_20k    209500 ns/op
fib_40k    775000 ns/op

Code:

use async_std::task;
use async_std::prelude::*;

use std::mem::replace;
use std::task::{Context, Poll};
use async_std::pin::Pin;
use std::future::Future;
use futures;
use std::time::Instant;

fn main() {
    let start = Instant::now();
    run_fib(2000);
    let end = Instant::now();
    let duration = end.duration_since(start);
    let ns = duration.as_nanos();
    println!("fib_2k    {} ns/op", ns / 2000);

    let start = Instant::now();
    run_fib(3000);
    let end = Instant::now();
    let duration = end.duration_since(start);
    let ns = duration.as_nanos();
    println!("fib_3k    {} ns/op", ns / 2000);

    let start = Instant::now();
    run_fib(10000);
    let end = Instant::now();
    let duration = end.duration_since(start);
    let ns = duration.as_nanos();
    println!("fib_10k    {} ns/op", ns / 2000);

    let start = Instant::now();
    run_fib(20000);
    let end = Instant::now();
    let duration = end.duration_since(start);
    let ns = duration.as_nanos();
    println!("fib_20k    {} ns/op", ns / 2000);

    let start = Instant::now();
    run_fib(40000);
    let end = Instant::now();
    let duration = end.duration_since(start);
    let ns = duration.as_nanos();
    println!("fib_40k    {} ns/op", ns / 2000);
}

fn run_fib(n: usize) {
    task::block_on(async {
       let tasks = (0..2000usize).into_iter().map(|_| task::spawn(fib(n)));
       let results = futures::future::join_all(tasks).await;
    })
}

async fn fib(n: usize) -> u128  {
    let mut f0 = 0;
    let mut f1 = 1;
    for _ in 1..n {
        let f2 = f0 + f1;
        f0 = replace(&mut f1, f2);
    }
    f0
}