Why is my engine kicking on every 3-5 minutes when the climate control is on?

First, 12.2 is low for a 12V battery. A fully charged 12V will be in the 12.7-12.9V range. However, this isn't causing your issue.

The engine will start whenever the car senses that the hybrid battery is low. If the car is sitting in Park it will run only long enough to bring the battery pack up to an acceptable minimum and then shut off. The A/C will bring the battery pack down to the minimum and the cycle repeats. If you drive the car the battery will charge to a higher level and therefore take longer to get to the minimum level when it's not charging. When I'm sitting in stop-and-go traffic my car will start every 3-5 minutes if the A/C is on.

In summary, I think your car is acting normally. You should look at the age of your 12V battery and either charge it or plan to replace it soon if it won't hold a higher charge. You might get a slightly longer run-stop cycle if the 12V is fully charged, since the charging load of the hybrid battery would be slightly reduced with a fully charged 12V,

 

the hybrid battery is simply showing its age

 

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How many miles are on the vehicle, and do you know if the HV battery is the original or a replacement?

An older HV battery won't have the same capacity as a new OEM battery.

You might try turning up the temp and or turning down the fan, both should save a tiny amount of juice.

You might find this post of interest : Prius camping

 

12.2 is marginally low. It should be around 12.8 and that .6 difference is important.

But this might be a sign that your HV battery is getting weak too.

 

The A/C compressor is variable speed. When little cooling is needed, it can run at around 250 watts. (Less demand than that, it can average even less, by cycling on and off). Increase the cooling demand and it can consume easily ten times that.

Makes quite a difference in how often the engine will run. Nothing in the description here sounds like it would concern me.