I didn't buy advance tickets for my trip, since I was advised if my flight was late it would be costly paying twice for rail tickets.
Now, I did a search for train tickets from Roma Termini to Siena and all the reservations that include an intercity fast train from Rome to Florence cost something like $140 Euros for two persons, at least twice the normal cost.
I guess this may be because of the crowds attending the funeral of the Pope.
This has nothing to do with the funeral of the Pope.
The "base" price for a ticket Rome to Florence is 55,- euro. The "ordinaria" price for a ticket from Firenze to Siena is 10.20.
(And you need to add the train from the airport. In fact, you should search Fiumicino - Siena). So you are indeed looking at 79,20 pp. for a fast route.
"base" and "ordinaria" are normal prices. They are the standard walk up fares that you pay if you buy your ticket at the last moment. If you buy in advance you get discounted prices.
What you could do is travel on regional trains, or an intercity. There are may options that are not that much slower then taking the High Speed train (because going via Florence is actually a big detour), and those are much cheaper.
You have the option to get two reseved bus tickets for fraction of the cost of the train that go directly from Rome Tiburtina Station to Siena for tomorrow on Flixbus:
https://www.flixbus.com
If Flixbus from Tiburtina is an option than going on regional trains all the way is a better one...
Just make sure to search "Fiumicino Aeroporte" to "Siena" on Trenitalia, as the cheap regional train routes do not go through Termini...
