This seems like a pretty significant bug (unless there was a ninja nerf to both summoner and scholar).

The spell speed from our gear does not affect Garuda / Eos / Selene's spell speed; neither does the trait "Enhanced Pet Actions" when it procs off a pet's critical hit. Both these things in the past would alter the pet's tool-tip for spells such as "Wind Blade" or "Embrace" to represent a cast time and recast time that was lower than their default values.

Time: Any
Frequency: Always
World name: Goblin
Character name: Any (Nurse Joy)
NPC name: Garuda-Egi, Eos, Selene.
Monster name: N.A.
Class/Level: Summoner / Scholar (50)
Party or solo: Both
In-game time: Any time
Area and coordinates: All areas
Steps:
1. Summon Garuda-Egi and note his "Wind Blade" cast and recast time.
2. Change gear to have significantly more Spell Speed (or significantly less).
3. Re-summon Garuda-Egi.
4. Observe that the cast and recast time for "Wind Blade" remains the same.

Alternatively:

1. Summon Garuda-Egi and note his "Wind Blade" cast and recast time.
2. Order Garuda-Egi to repeatedly cast "Wind Blade" on a Training Dummy.
3. Wait until the trait "Enhanced Pet Actions" occurs off of a "Wind Blade" critical hit.
4. Observe that the cast and recast time for "Wind Blade" remains the same.