What breed was Lawrence of horse?

There's no record of Lawrence of Arabia having a horse named "Lawrence". It's likely you're thinking of the famous British military officer T.E. Lawrence, also known as Lawrence of Arabia.

While he was known for his equestrian skills and used horses extensively during his time in the Middle East, he didn't have a specific named horse.

He rode various horses during his service in the Arab Revolt, and it was these horses that were famously depicted in the film "Lawrence of Arabia", not a single named horse.