Special Travel Pace (DMG p. 242–243):
- In 1 hour, you can move a number of miles equal to your speed divided by 10.
- For a fast pace, increase the rate of travel by one-third.
- For a slow pace, multiply the rate by two-thirds.
For each size category above Medium, double the creature's carrying capacity and the amount it can push, drag, or lift. For a Tiny creature, halve these weights.
So a Riding horse has a Strength score of 16, is Large so that's up to 160 pounds (16 × 5 = 80 then 80 × 2 = 160), not slowed. It has a speed of 60 feet (or 6 miles an hour) when traveling at a normal pace. This is increased by one-third when traveling at a fast pace; 6 × 1⅓ = 8.
You can travel 8 hours a day without exhaustion. Plus you can gallop (inside that 8-hour limit) at double the normal pace (6 × 2 = 12) for 1 hour. It makes the most sense to go fast for 7 hours (7 × 8 = 56), then gallop the last hour (1 × 12 = 12), for a total of 56 + 12 = 68 miles a day.
That's 226.66% the distance that an "unslowed" human party could travel on foot at a "fast pace" (30 miles in the same time period). That's equal to a fifth of the width of Florida! To do the math, this horse could cross the whole USA in under 31 days!)
Is this right?
The rounding rule is just about direction of rounding. Normal math rules are otherwise used—which means, don't round until the whole calculation is finished. (Rounding after each step in multi-step math calculation is verboten by the normal rules of math because it causes errors.) Since the only change to normal math rules 5e makes is which direction to round, normal math rules apply to 5e rules in all other ways.
So at the very least, in 8 hours, 8 x 7 = 56, plus the one-hour gallop for 12 more, total 68? if a horse isn't limited to the same movement as that of humans with a 30-foot speed listed in PHB p. 182, which RAW, other than galloping once per "long" rest (I'm reading between the lines): IT IS.
MAJOR ADDITIONAL INFO
XGtE (Xanathar's Guide to Everything) page 80, cobblers tools note: must have cobblers tools and be proficient with them
Maintain Shoes. As part of a long rest, you can repair your companions shoes. For the next 24 hours, up to six creatures of your choice who wear shoes you worked on can travel up to 10 hours a day without making saving throws to avoid exhaustion.
so 2 extra hours travel or up to 8 EXTRA miles, if you wear PC race footwear, so NOT A HORSE (38 miles total, beating horses RAW of 34 miles)