If a wire gets warm then (IMO) it is too small. That is one of the tests for electrics - high resistance from bad contacts and undersized conductors results in heat (p = i*i*R).
In general for 12V LEDs
the olde div-10 sizing rule should suffice - ie Wattage divided by 10 to give Amps - and then choose
at least the next sized cable.
Otherwise use the LED's max current to determine cable size noting that that max may be at its highest or lowest operating voltage (see below).
FYI...
Cable ratings may be to prevent insulation breakdown in which case warmth & relatively high resistance (smaller diameter) may be acceptable.
Or cable ratings may be limit voltage drops (per length) within acceptable limits. IE - a
thin cable may be fine temperature wise but result in too low a voltage being available at the load.
I suspect many cabling problems with high-power automotive LEDs result from cable sizing assuming 12V or 12.8V or 14.2V etc without realising they may be
constant power loads whose cabling should be sized using the lowest voltage.
EG - a 27W LED that operates down to 9V will draw 3A @ 9V plus maybe a bit for inefficiency. [ie, does the LED itself use 27W or is that the entire LED module's rating? If the LED itself is 27W then its total input is closer to say 30W assuming 90% converter/ballast efficiency.]
Assuming 27W total load, that means 3A @ 9V compared to 2.1A @ 12.8V or 1.9A @ 14.2V.
{Note that high power/current LEDs usually have electronic drivers aka ballasts or converters; they rarely rely on mere resistors as do low power 20mA & 50mA etc LEDs. Tho such drivers were typically step-down aka buck converters with constant current (else voltage) outputs, these days buck/boost converters are often used since that's cheaper than using lower output voltages. EG, a 12V LED may be a 24V LED string since one converter with one 24V string is cheaper than 2 converters with two 12V strings. As a bonus, that "LED" can be used for 12V & 24V systems.}
I suspect too that because high-power LED output does not depend on input voltage - ie, wiring voltage drops are not as critical as they were for halogens - manufacturers have gone for thinner cables but have overlooked the thermal issues.