Is there a way to powerlevel your chocobo companion?

[Kirika-]

Treats do give the Chocobo a set exp under the Train option at the Stables, there are restrictions:
Only the Chocobo's owner can stable it or give it Color Change foods, however if it's in a FC stable, the FC members can Train it
There is a hour cool down on training, you can't stable it and then repeatedly Train it
The Condition of the stable directly effects the amount of experience gained
If you Train the Chocobo with Curiel Root, not only does the Chocobo gain experience, but it will also give the Chocobo an experience gain buff, which is more effective once it becomes the Chocobo's favorite treat
https://ffxiv.consolegameswiki.com/wiki/Chocobo_Raising

If you had stabled the Chocobo for the color change, maybe one or more of your FC mates decided to keep up a Training regiment while your were away, resulting in a lot of experience for it.

As for "power leveling", probably the fastest way is via FATEs that are the type where you have to kill many monsters.
As you sync to the FATE's level, so does your Chocobo, so you are going to be getting very good exp per kill for both you and your Chocobo (or just your Chocobo if you are already capped)
Just find FATEs (any level) that you can blast through easily and have your Chocobo with you.
Also, weekly, the Achievements/Challenge Log gained for kills within you and your Chocobo's level that you are both participants in is a massive experience reward to your Chocobo.
https://ffxiv.gamerescape.com/wiki/B..._I_(Challenge)
https://ffxiv.gamerescape.com/wiki/B...II_(Challenge)

Each time your Chocobo ranks up, it gets Skill Points equal to the Rank to spend in the Skill Tree, so at Rank 1 he gets one SP, Rank 2 he gets 2, and so on up to Rank 10.
Once your Chocobo hits Rank 10, you will need to feed him a Thavnairian Onion in order to unlock the ability to gain experience for the next rank. https://ffxiv.gamerescape.com/wiki/Thavnairian_Onion
From Rank 10 on, the amount of points earned per Rank up is:
10 SP is earned for each level 10 - 16
11 SP for level 17
12 SP for level 18
13 SP for level 19
14 SP for level 20

Max rank is Rank 20 and at which the Chocobo will have earned enough SP to "buy" all Skills.

[Omedon]

Treats do give the Chocobo a set exp under the Train option at the Stables, there are restrictions:
Only the Chocobo's owner can stable it or give it Color Change foods, however if it's in a FC stable, the FC members can Train it
There is a hour cool down on training, you can't stable it and then repeatedly Train it
The Condition of the stable directly effects the amount of experience gained
If you Train the Chocobo with Curiel Root, not only does the Chocobo gain experience, but it will also give the Chocobo an experience gain buff, which is more effective once it becomes the Chocobo's favorite treat
https://ffxiv.consolegameswiki.com/wiki/Chocobo_Raising

If you had stabled the Chocobo for the color change, maybe one or more of your FC mates decided to keep up a Training regiment while your were away, resulting in a lot of experience for it.

As for "power leveling", probably the fastest way is via FATEs that are the type where you have to kill many monsters.
As you sync to the FATE's level, so does your Chocobo, so you are going to be getting very good exp per kill for both you and your Chocobo (or just your Chocobo if you are already capped)
Just find FATEs (any level) that you can blast through easily and have your Chocobo with you.
Also, weekly, the Achievements gained for kills within you and your Chocobo's level that you are both participants in is a massive experience reward to your Chocobo.

Each time your Chocobo ranks up, it gets Skill Points equal to the Rank to spend in the Skill Tree, so at Rank 1 he gets one SP, Rank 2 he gets 2, and so on up to Rank 10.
Once your Chocobo hits Rank 10, you will need to feed him a Thavnairian Onion in order to unlock the ability to gain experience for the next rank. https://ffxiv.gamerescape.com/wiki/Thavnairian_Onion
From Rank 10 on, the amount of points earned per Rank up is:
10 SP is earned for each level 10 - 16
11 SP for level 17
12 SP for level 18
13 SP for level 19
14 SP for level 20

Max rank is Rank 20 and at which the Chocobo will have earned enough SP to "buy" all Skills.

This was very helpful, thank you! I also didn't know that the SP per level scaled up!

[StouterTaru]

Only the Chocobo's owner can stable it or give it Color Change foods, however if it's in a FC stable, the FC members can Train it
There is a hour cool down on training, you can't stable it and then repeatedly Train it

There are two separate 1-hour timers for stable feeding, so your chocobo can be fed twice per hour. One timer is for you personally, the other timer is for everyone else. Make an alt character and and have it join your FC, and you can double feed.