There is a scene in Captain America: Civil War where Steve Rogers asks Tony about his relationship with Pepper and Tony Stark says that they are taking a break. So did they break up simply because Pepper couldn't take the fact of him continuing to be Iron Man?
He's "hard to deal with".
In the film, Tony is fairly candid with Steve Rogers as to the reasons for the strain in his relationship with Pepper. He explains to Steve that:
The relationship had begun to unravel after the climactic events of Iron Man 3 in which Pepper had been almost killed.
His continued use of Iron suits despite Pepper's concerns for his safety had driven a wedge between them.
The Sokovia incident, which was a product of Tony's recklessness (via Ultron), had worsened his already-strained relations with Pepper.
He then admitted that, on top of these specific obstacles, he is simply "hard to deal with".