I have a number of transformers and would like to experiement using then
at
a higher frequency....let say 500 hz and / or higher.....

Would they still work?...I am assuming so...

Would they run cooler?....

Could anyone foresee any negative effects?....
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Running transformers on a higher freq is fine provided the core can manage
the higher freq.

During the valve era ,aeronautical radio equipment ran on 400 Hz resulting
in smaller and hence lower weight transformers.

However running power supply transformers at an AC frequency lower than the
one they are designed for might cause the laminated core to overheat.

In practice : 50 Hz transformers meant for use in Europe and elsewhere can
be readily used in the US and Canada.
However transformers designed for 60 Hz having 'marginal cores' could
overheat when used on a 50 Hz supply.
So equipment with 'wallwarts' purchased in the US ,(apart from being
suitable for 117 V only ) I never use on a 50 Hz supply system.

Nowadays with much equipment having a switch mode power supply , the
frequency / voltage problem no longer exists.

Frank GM0CSZ / KN6WH