Wakefield 120. The basic, plain-yogurt, of thermal grease, and only a little less appetizing. (No please, don't actually eat it..)
Thermal conductivity doesn't matter much, because a properly prepared joint has very little air gap to fill, and excess grease squidges out, leaving a minimum height joint. The transistor (RthJC), insulator (if applicable) and heatsink dominate overall thermal resistance.
Perspective shift: I would say If grease is adding notable resistance, the joint is wrong, not the grease.
AFAIK, lithium grease is a fairly ordinary petroleum or synthetic oil, gelled with a lithium soap (a low-solubility variant of the active ingredient in regular bar soap; basically, soap scum). The soap particles partially dissolve, thickening the oil, but also disperse, remaining in suspension, forming a gel that stays in place, and giving something solid but non-abrasive for the bearing surfaces to rest on, when not in (hydrodynamic) motion.
Thermal grease is made much the same way, of course it isn't used under relative motion, heh. As a result, more abrasive materials can be used as the filler; ZnO is typical. (Indeed, wiping thermal grease off a bare aluminum heatsink, you'll find it leaving dark streaks, as the particles abrade and polish the metal.) A silicone oil is usually used, being chemically inert, and perhaps better at staying in place. Still, a common failure mode -- whether by improper formulation, application, overheating, thermal cycling, or
it just does that -- is separation, leaving a dry powdery joint under the device, which needs to be cleaned and reapplied.
So, yes indeed -- it has a similar appearance, for similar reasons, but using different materials.
Anyway, high-K grease: the trouble is, this can only be done two ways: using higher-K filler, and using less oil. You can very well end up worse off, because the grease doesn't squeeze out, and now you have, sure it's 15 W/(m.K) or whatever, 20 times better than the plain white stuff, but maybe the white stuff crushes down to 20µm average, while you've got a 0.5mm thick blob under the device, and it's not even fully spread out, maybe the contact area is half. Or maybe the clamping force is so great, and uneven, that the device is cracked before it can sink any power at all(!). Applying those super-thick compounds is nontrivial.
But, combining all of the above: if you're in a situation where the joints just aren't flat, there's big gaps, the faces are rounded, whatever, nothing fits right, and it's not practical to fix the surfaces; that might be a good place to use a higher-K grease, instead of the low-K stuff that'll end up filling large pockets and adding significant Rth.
Soft rubber gap pads might also be attractive: these are available in fairly high K nowadays too (5 and up), and the softness means they fill voids very well. An excellent application is using them to clamp a PCB, components and all: they can fill in the gaps between SMT components, making contact all around, as well as to the board itself, vastly increasing power dissipation. Downside: they are solid rubber products still, not puttylike (well, some of them maybe?), so start to crack and tear when strained too far; they aren't great for rigid (screw and strong spring-clamp) mounting. On the upside, being soft rubber materials with some surface tack, they need hardly any holding force, and a soft spring clamp can do the job. Self-adhesive (firmer, very tacky) materials are also available.
Tim