The Gnome Saboteur's Chameleon Mutagen gives them a climb speed = to 1/2 their base speed. So an increase to base speed would also increase their climb speed by half.
However, Expeditious Retreat specifically says it does not effect alternative modes of travel such as Fly, Swim, Climb, etc. but it seems that it is talking about natural travel abilities.
RAW, would Expeditious Retreat increase the Gnome Saboteur's climb speed from their Chameleon Mutagen ability?
The spell expeditious retreat, which can't increase a climb speed, increases base land speed. The gnome saboteur's chameleon mutagen grants a climb speed of half the gnome saboteur's base speed.
The spell expeditious retreat [trans] (Pathfinder Role-playing Game Core Rulebook 279) "increases your base land speed by 30 ft." Here's the definition of base land speed:
Base land speed is your unencumbered speed. Base land speed is calculated by applying all modifiers to your character’s speed with the exception of armor or encumbrance adjustments or any effect that modifies your "normal speed".
Normal speed, then, is defined as follows:
Your normal speed is your total encumbered speed (if any encumbrance applies). Normal speed is calculated by applying any armor or encumbrance reduction as indicated on Table: Armor and Encumbrance for Other Base Speeds, to your base speed.
The chameleon mutagen of the saboteur achetype is "granting him a climb speed equal to half his base speed, for 10 minutes per saboteur level."
Base speed, then, is defined as follows:
Base speed is your unencumbered speed for a specified type of movement. Your base speed for any movement type is calculated in a similar manner as described in Base Land speed. When a speed type is not referenced, base speed usually implies base land speed.
The spell expeditious retreat explicitly "has no effect on other forms of movement, such as climb, fly, or swim," but it explicitly affects base land speed, which is, for most, also base speed. When the gnome saboteur chugs a chameleon mutagen, the mutagen looks only at the base speed. When the gnome saboteur uses a wand of expeditious retreat, the spell looks only at the gnome saboteur's base land speed.
That the two happen to be awesome together is a happy accident.
The spell expeditious retreat "has no effect on other forms of movement, such as climb, fly, or swim" so while that spell's on, that spell can't change any other speed no matter what. Done. Full stop. Because the spell says so. That's why. "But...," you say. Don't care. The spell says no effect. If you've another form of movement beside a land speed, expeditious retreat has no effect. No effect, okay? None. Nuh-uh. No way. Stop trying. Mutagen? Magic item? Fly speed twice your base speed because a god gave it to you? The spell expeditious retreat just don't care, man. Let it go.
Seriously. That's your best bet.
(By the way, apparently the various definitions of speed in Pathfinder are complex enough to warrant a guide.)
As per the rules, NO
The chameleon mutagen does not give you a new ability, it enhances a mode of movement you already have. If anyone can climb at 1/4 of its normal speed, the mutagen enhances it to 1/2 your normal speed, but you are still climbing.
Expeditious Retreat specifically states that it does not affect climbing. Aditionally, both climbing and the mutagen states that your climbing speed is calculated from you base or normal speed (emphasis mine):
From climb page:
...With a successful Climb check, you can advance up, down, or across a slope, wall, or other steep incline (or even across a ceiling, provided it has handholds) at one-quarter your normal speed...
From chamaleon mutagen:
...This grants him a circumstance bonus on Stealth checks equal to half his alchemist level (minimum +1), as well as granting him a climb speed equal to half his base speed, for 10 minutes per saboteur level...
This would exclude bonuses, and the spell states that the speed increase is a enhancement bonus:
This spell increases your base land speed by 30 feet. This adjustment is treated as an enhancement bonus.
So expeditious retreat has no effect on you while climbing.
By giving you a climb speed, it gives you a +8 bonus to your Climb skill, not just raising the speed by which you can climb in relation to the normal skill use.
– NinthNova Jan 15 '14 at 04:46