The two rules elements are quoted below. There are three similar questions, but they don't seem to address my specific question:
- What is a Centaur Thief's climbing speed?
This suggests the Thief would negate Equine Bond, but with Tumbling Fool you just gain a Climbing speed equal to your current speed. I don't think that is the same effect?
- Is there a functional difference between having a listed climb speed and being able to climb at full walking speed?
This also deals with the same effect as the first question, but also dives into how items interact.
- How does the Athlete feat affect the Centaur playable race?
Finally, this also deals with ignoring extra movement.
What is your climbing speed and what restrictions do you have with Equine Build and Tumbling Fool?
Or more precisely, does the ability of "climbing speed equal to your current speed" also work the same as "no longer costs you extra" effects?
The Rules text:
The updated Centaur on page 9 of Mordenkainen Presents: Monsters of the Multiverse has an ability called Equine Build:
You count as one size larger when determining your carrying capacity and the weight you can push or drag.
In addition, any climb that requires hands and feet is especially difficult for you because of your equine legs. When you make such a climb, each foot of movement costs you 4 extra feet instead of the normal 1 extra foot.
The 3rd level ability Tumbling Fool has these rules:
At 3rd level, you master a variety of acrobatic techniques that allow you to evade danger. As a bonus action, you can tumble. When you tumble, you gain the following benefits for the rest of your turn:
• You gain the benefits of taking the Dash and Disengage actions.
• You gain a climbing speed equal to your current speed.
• You take half damage from falling.