Fast Train from Rome

What time of day are you wanting to travel?

When? I looked at Thursday, and there's a direct train at 1157 taking 2 hr 56 min. But there's an earlier train at 0910 taking 2 hr 41 min but requiring a change in Florence.

Knowing the date of travel would help.

Without checking, I don't think there is a "fast train" from Rome to Pisa. Probably dealing with ICs and Regionale Just checked. Mostly about three hours give or take 15 min or so with one change. There are a couple of ICs that go direct but still around 3 houras.

A direct, no changes train at 9:57 is an IC 510 train and gets to Pisa in 3 hours and 2 minutes.
There are two earlier trains at 7:35 and 8:35 that are combos of Frecciarossa changing to a Regionale Veloce at Firenze SM Novella, and get to Pisa in 2 hours and 53 minutes.

There are other similar trains later in the day; that's why I'm asking what time of day you want to travel. Specific day would be helpful for drilling down to departure times and train #s, and also if you'd prefer a no-changes journey over one with a change & saving just a few minutes.

Editing to add: the direct, no changes 11:57 that CJean noted is a Frecciabianca. It's also, for the random date I'm using, one of the more economically priced tickets.
The 9:10 noted (requires a change) is a Frecciarossa + Regionale Veloce combo

You can also check out Italo (https://www.italotreno.com/en). Looks like similar travel times, change trains in Florence. Italo competes with Trenitalia. I've ridden both, both are excellent trains.