The Acrobatics section of the Core Rulebook PRD currently says,
In addition, you can move through a threatened square without provoking an attack of opportunity from an enemy by using Acrobatics. When moving in this way, you move at half speed. You can move at full speed by increasing the DC of the check by 10. You cannot use Acrobatics to move past foes if your speed is reduced due to carrying a medium or heavy load or wearing medium or heavy armor. If an ability allows you to move at full speed under such conditions, you can use Acrobatics to move past foes. You can use Acrobatics in this way while prone, but doing so requires a full-round action to move 5 feet, and the DC is increased by 5. If you attempt to move through an enemy's space and fail the check, you lose the move action and provoke an attack of opportunity.
Therefore, the answer is half speed during the entire movement.
The DCs are given in a table that I did not copy.
The first example. You only move at half speed while you're avoiding provoking the AoO. Once you're in squares that aren't threatened you should be free to go back to full speed movement.
It's only half speed while avoiding. The rest of your movement s fine.
Mathmuse quoted the rule, but I think misapplied it.
"you can move through a threatened square...when moving in this way you move at half speed..."
You only move half speed when moving through threatened squares.
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This FAQ includes the text "She must move at half speed while threatened by these foes and can choose which to check against first." To me, that suggests that only the threatened squares require slowed movement. The term "half speed" is rather confusing in this case, though.
Mathmuse quoted the rule, but I think misapplied it.
"you can move through a threatened square...when moving in this way you move at half speed..."
You only move half speed when moving through threatened squares.
The line from the Acrobatics FAQ that Kalindlara found, "She must move at half speed while threatened by these foes ...," shows that I was wrong.
I had two reasons for my interpretation, neither of which are rigorous.
First, that is the way I played Tumble skill in D&D 3rd Edition. However, checking the Tumble rules at d20srd.org, I see that it says, "Tumble at one-half speed as part of normal movement, provoking no attacks of opportunity while doing so." The phrase "part of" implies that tumbling movement can be combined with normal movement.
Second, movement restricted to particular squares usually falls under the Hampered Movement rules, that refer to specific squares requiring double or triple movement rather than referring to half or one-third speed. If someone used tumbling only on threatened squares, then calculating changing speeds would be a very awkward way to track the maximum movement allowed.
In most other examples of another movement speed, such as sneaking or running, the person usually wants to continue that kind of movement for the entire move action, so the issue of mixed speeds seldom occurs. Mixed movement speeds happened a few times in my games, such as climbing a cliff and having movement left over to walk on the clifftop, or walking to a river and swimming across it. I had to split the movement into percentages, "You used 67% of your movement climbing, so you have 33% left for walking, which would be 10 feet." The calculations slow down gameplay.
Nevertheless, perhaps the designers viewed tumbling as a type of movement, like walking, running, climbing, swimming, flying, or burrowing, and always use speed for types of movement, regardless of the inconvenience. In actual gameplay, the GM and player would switch to counting the tumbled squares as double movement to make the rule manageable.
I worry, however, that this rule will also let the character combine tumbling with running, to avoid that awkward loss of Dexterity bonus to AC from running. Even if the character fails the Acrobatics check, the character would be tumbling rather than running during the attack of opportunity, so he or she would have the full Dexterity bonus to AC.
Well, Pathfinder accidentally nerfed tumbling when they changed its DC to be the attacker's CMD, so making tumbling more versatile would help correct that.