I'm going to answer this from my experience with laser cutters. This probably means I'll get down votes, as being outside of the Star Wars universe, but whatever.
Laser beams in and of themselves aren't hot. It's the interaction between the laser and other materials that makes heat. In a typical laser cutter (specifically non-fiber optic lasers), there are mirrors that redirect the laser beam from the laser emitter to the intended target. These mirrors take the whole brunt of the laser beam without any cooling. There's also a focusing lens that the beam passes through to go from a "thick" laser beam to one that looks like an hourglass, with the focus being the skinny part of the hourglass and where you generally try to get that point at the top of your material to cut or etch. This lens does have some cooling, since often there's usually a gas being directed at the piece being cut for various possible reasons, which I won't get into here. This gas stream isn't intended for cooling, though.
The lens and the mirrors can get hot, but only when they get dirty or are used with a beam that's beyond their intended strength. This is because they are intended to either reflect or refract the light, not absorb it. It's in the best interest of the cutter operator to keep these clean, so they don't get damaged or reduce the effect of the laser.
The laser source itself gets hot and needs cooling, yes, but it also has large amounts of electricity flowing through it to excite the particles to create the laser beam. It also interacts with the light differently, since it's trying to contain the light and get it to exit in only a single direction. Because of this, I'd wonder how a light saber user could hold than handle, rather than worry about the blade. Creating enough laser to cut through a massive steel door is going to create an equally massive amount of heat. It takes over 150 watts for a laser to cut through even thin sheet steel.
I've seen YouTube videos of 4000+ watt lasers cut through 1"-4" of steel, which is impressive enough, and that's a beam the size of a pencil lead, or thinner. You need much more power (and heat generated) to have a 1"-2" laser to cut through a blast door, like we see Qui-Gon and others do.
Laser interactions and cooling
Then there's also the casing around the focusing lens. To direct the gases in the correct direction for the cut, there's a cone shaped nozzle that also contains the laser beam. The only time I've noticed these getting hot is when the beam isn't focused correctly and hits the side of this cone. If it passes by without touching, no heat is transferred to the cone. And this heating happens regardless of the gases blowing through it, so the cooling effect of that gas is very minimal with respect to the heat generated by the laser beam actually interacting with a material.
If you look at the example image below, the laser enters the opening at the top left, then it immediately hits a mirror contained in the 45 degree angle section directing it straight down (if focused properly). Right at the knurled section is where the lens is situated, and it's knurled so you can take this assembly apart and either replace or clean the lens. The barbed tube near the bottom right is the gas (generally air) inlet. The bottom is another opening where the gas and the laser exit. If there wasn't a lens in this, you could blow through it without any problems, so there are no interactions between it and the laser beam except for the mirror and lens. (All the knobs are simply adjustment screws.)
Also, air is a really poor conductor of heat. It's so poor, that it's often used as an insulator. How else can you cook over a hot stove and not get burned, except when in accidental direct contact with the stove, pan, or a cooking utensil. You can hold your hand only a few inches over a hot burner and not have a problem, but it'll burn your food with no problems, due to direct contact between good heat conductors and your food.
Air is a collection of gases, and it is not a good conductor or radiator. Air is excellent at convection, but the amount of heat that can be transferred is minimal because the low mass of the substance cannot store a great deal of heat. Air is used as an insulator in coolers and building walls.
Also, I've done a variety of organic materials in the laser cutter, which takes very little laser power to cut. An 80 watt laser cuts through a single slice of bread almost as if it wasn't there. It also etches graham crackers, marshmallows, and chocolate pretty easily. (Yes, I etched the components of a S'mores.) I've also talked with someone who etched hamburgers. In fact, I've been meaning to do a video on how dangerous laser cutter are, by using hot-dogs as a substitute for fingers.
On the flip side, I've etched and cut paper and corrugated cardboard without setting it on fire. All this is done by adjusting the power and speed of the machine, so you put in only the amount of laser into the material you need to vaporize the material, and not heat up the surrounding material.
So what does any of this mean to a light saber? Well, it shows that a tool that can cut through a thick steel door can easily pass through even the torso of a human with little to no resistance. It also shows that a cauterizing effect can be possible, due to only "minor" heating of the material surrounding the cut, rather than just setting everything on fire.
It also shows that the laser beam itself isn't really much of a problem, but the hand grip should be a massive heat problem. Of course, getting a laser strong enough to cut through thick steel out of a tiny tool like that can be hand-waved by "alien/advanced engineering", so the heat sink/dissipation problem could be solved the same way.
So far, I've only dealt with laser cutters that are 100 watt or less, so it would be interesting to hear from someone experienced in larger industrial lasers.
For those of you who don't think a light saber is a laser weapon, Wookieepedia states them as also being called laser swords, and that it was designed after learning how to "freeze" a laser. The article states that people unfamiliar with the weapon called it a laser sword, but Luke Skywalker called it that in "The Last Jedi". You can say he was being sarcastic about it's use, but he still did say it.
The lightsaber, also referred to as a laser sword by those who were unfamiliar with it, was a distinctive weapon, the very image of which was inextricably bound with the mythos of the Jedi Order and their polar opposites, the Sith.
The first lightsabers came into being when the precursor Je'daii Order combined advanced offworld technology with a forging ritual, learning how to "freeze" a laser beam.
Wikipedia states that it uses the same crystals used to create the superlaser for the Deathstar.
The source of a lightsaber's power is a kyber crystal. These crystals are also the power source of the Death Star's superlaser.
Laser can be used to generate plasma, and high power enough lasers do generate plasma.
Filamentation also refers to the self-focusing of a high power laser pulse. At high powers, the nonlinear part of the index of refraction becomes important and causes a higher index of refraction in the center of the laser beam, where the laser is brighter than at the edges, causing a feedback that focuses the laser even more. The tighter focused laser has a higher peak brightness (irradiance) that forms a plasma. The plasma has an index of refraction lower than one, and causes a defocusing of the laser beam. The interplay of the focusing index of refraction, and the defocusing plasma makes the formation of a long filament of plasma that can be micrometers to kilometers in length. One interesting aspect of the filamentation generated plasma is the relatively low ion density due to defocusing effects of the ionized electrons.
Other publications also support plasma being created by lasers, and lots of research has been done on the topic.
The interaction between a pulsed laser beam and any substance is extremely complex . It is a non-linear process, dependent upon laser characteristics (fluence, pulse rise-time and duration, wavelength, beam quality), substrate composition and surface character, and the environment in which the plasma forms (pressure and composition). We counted more than 1500 publications in the last 5 years on laser-induced plasmas related to LIBS, many having to do with studies of the influence of laser wavelength, [...]