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I am currently creating a cat-like creature that lives in the trees. It is a diurnal ambush predator that's a little over a foot (30 cm) in height and two feet (60 cm) in length (not including tail). The average temperature is in the 90s Fahrenheit / around 35 Celsius.

How big would its horns have to be to help it cool off? And what would be the best shape for them?

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    $\begingroup$ Horns aren't ideal for cooling; you want to maximize the ratio of surface area to volume. Ears are a much better options. You might want to look at fennecs and elephants. $\endgroup$ – Matthew Apr 12 at 13:01
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    $\begingroup$ Here's the only scholarly article I could find related to your question. The TLDR is if horns are the only cooling mechanism, they'd have to be pretty big. jstor.org/stable/42901121?seq=1 $\endgroup$ – Andrew Brēza Apr 12 at 13:45
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    $\begingroup$ P.s. Horns and tree-dwelling are not good friends. $\endgroup$ – PcMan Apr 12 at 15:51
  • $\begingroup$ Also struggling with the concept of a passive/radiator type cooling system, when ambient air is 35 Celcius. At best that's about 4 degrees below a cat's body temp. $\endgroup$ – PcMan Apr 12 at 16:40
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As the comments pointed out, horns aren't very ideal as a cooling device for a fair few reasons.

Horns are nowhere near as efficient at cooling a creature as opposed to ears for example. As ears have a constant flow of blood, the blood can carry the heat directly to the thin outer edge with the biggest surface area and airflow, where it quickly gets dissipated. Horns don't have a constant blood flow in them, unless you have the velvet-covered horns deer have, which can get caught in branches and tear easily. Besides, bone is thick, so it has less surface area and airflow, so retains more heat naturally. You would need a tree like structured horn, with thick base to carry the heat, into smaller branches with more airflow and surface area. The outcrops would need to be thin, and hence won't be very strong. If you would need them to be strong, you would need a lot of surface area still, making the horns unnecessarily big and unwieldy.

All in all, the horns would be big, velvet covered, and tree-like. A lot of disadvantages for something like a cat-like predator, and don't make sense from an evolutionary standpoint as there are a plethora of better options available.

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  • $\begingroup$ WRT deer horns, they're only in velvet part of the year, so not useful for cooling at other times. $\endgroup$ – jamesqf Apr 12 at 16:25
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JUST COPY THE NORMAL CATS

It doesn't make much biological sense to be honest...

The horns on animals are made of the same materials as nails and are practically dead tissue. That's the main reason why animals don't bother much once their horns are broken off or fall off. The only living part of the horn is actually beneath the skin, the one that holds it in place and grows it out as time goes on.

Since you pointed out you're creating cat-like creatures, it would be proper for them to have the same cooling system cats have, which is cooling themselves through their tongue and through their paws (their sweat glands are located on their foot pads).

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