Hi FLFisherman,
Welcome to the Mattress Forum! 
There is no meaningful information about this mattress listed on the Mattress Firm site, which isn’t a surprise, as neither Mattress Firm nor Simmons are known for providing specific details about the componentry used within their mattresses.
When selecting any mattress, I would always make sure that you find out the information listed here so you can compare the quality of the materials and components to the durability guidelines here to make sure there are no lower quality materials or weak links in a mattress that would be a cause for concern relative to the durability and useful life of a mattress before making any purchase.
The Aubrie Firm is shown to have:
1" BlackIce HD memory foam
4.5" Black HD memory foam
1" Energy polyfoam
7" Beautyrest Black Support polyfoam core
There are no densities provided to make an educated analysis of this product.
Unfortunately, this is common with many of the major brands such as Sealy/Stearns & Foster, Simmons, and Serta, as they all tend to use lower quality and less durable materials in their mattresses than most of their smaller competitors that will tend to soften or break down prematurely relative to the price you pay which is why I would generally suggest avoiding all of them completely (along with the major retailers that focus on them as well) regardless of how they may feel in a showroom along with any mattress where you aren’t able to find out the type and quality/durability of the materials inside it (see the guidelines here along with post #3 here and post #12 here and post #404 here).
Classic Brands also doesn’t list any meaningful information about this mattress on the amazon page, and I’m not sure of the accuracy of the little information they supply there. Apparently there is 2.5" of gel memory foam in the upper layer, followed by a total of 3" of polyfoam that has some gel added to it, and beneath that 6.5" of a polyfoam base (erroneously listed as memory foam on amazon). No densities of the foams are provided on the amazon site or the Classic Brands site. On amazon it claims the memory foam is CeritPUR-US certified.
You can read more about the Dormia / Classic Sleep bankruptcies and their purchase by their supplier and their change into Classic Brands in this article and a more detailed history yet in this article. A forum search on Classic Brands (you can just click the link) will also bring up more comments and feedback about them as well.
Their factory in China currently manufactures mattresses under several different names and makes private label mattresses as well. There are some comments about Chinese mattresses that are shipped compressed from overseas in post #6 here that would be worth reading.
Overall, I would be very cautious about buying any mattress where you can’t find out the specifics of the materials inside it so you can make sure there are no weak links in the mattress and make more meaningful comparisons to other mattresses. Outside of how well a mattress matches your specific needs and preferences in terms of PPP (Posture and alignment, Pressure relief, and your own Personal preferences), a mattress is only as good as the quality of the materials inside it regardless of the name of the manufacturer. If you are able to find out the type and quality of the materials in either of these mattresses and you list them on the forum I’d be happy to make some comments about the quality of the materials.
Regarding cooling memory foam claims, you can read more about phase change materials in post #9 here and at the end of post #4 here) and you can read more about the various different types of gel foams in post #2 here. In general terms gel foams will tend to have a temporary effect on temperature while you are first going to sleep until temperatures equalize but have less effect on temperature regulation throughout the course of the night.
Without knowing the specific within a mattress, you can’t make educated comparisons. There is more information in post #9 here about the different ways that one mattress can “match” or “approximate” another one. Every layer and component in a mattress (including the cover and any quilting materials) will affect the feel and performance of every other layer and component and the mattress “as a whole” so unless you are able to find another mattress that uses exactly the same type of materials, components, cover and quilting, layer thicknesses, layer firmnesses, and overall design (which would be fairly unlikely) then there really isn’t a reliable way to match one mattress to another one in terms of “comfort”, firmness, and PPP based on the specifications of the mattresses (even assuming that you can find out all the specifications you would need for both mattresses you are comparing in the first place).
There are also no “standard” definitions or consensus of opinions for firmness ratings and different manufacturers can rate their mattresses very differently than others so a mattress that one manufacturer rates as being a specific firmness could be rated very differently by another manufacturer. Different people can also have very different perceptions of firmness and softness compared to others as well and a mattress that feels firm for one person can feel like “medium” for someone else or even “soft” for someone else (or vice versa) depending on their body type, sleeping style, physiology, their frame of reference based on what they are used to, and their individual sensitivity and perceptions. There are also different types of firmness and softness that different people may be sensitive to that can affect how they “rate” a mattress as well (see post #15 here) so different people can also have very different opinions on how two mattresses compare in terms of firmness and some people may rate one mattress as being firmer than another and someone else may rate them the other way around. This is all relative and very subjective and is as much an art as a science.
In other words … the short version of all of this and the “bottom line” is that spending time testing major brand mattresses or any mattress where you can’t find out the specifics of the materials and components inside it and where another manufacturer doesn’t make a better quality/value mattress that they specifically describe as being similar is mostly wasted because it would be too risky to purchase and it can’t be used as a reference point to purchase another mattress that is “similar”.
I’m not sure what you’ve read since you found the site but just in case you haven’t read it yet … the first place to start your research is the mattress shopping tutorial here which includes all the basic information, steps, and guidelines that can help you make the best possible choice … and perhaps more importantly know how and why to avoid the worst ones.
Two of the most important links in the tutorial that I would especially make sure you’ve read are post #2 here which has more about the different ways to choose a suitable mattress (either locally or online) that is the best “match” for you in terms of “comfort”, firmness, and PPP that can help you assess and minimize the risks of making a choice that doesn’t turn out as well as you hoped for and post #13 here which has more about the most important parts of the “value” of a mattress purchase which can help you make more meaningful quality/value comparisons between mattresses in terms of suitability (how well you will sleep), durability (how long you will sleep well), and the overall value of a mattress compared to your other finalists based on all the parts of your personal value equation that are most important to you (including the price of course and the options you have available after a purchase if your choice doesn’t turn out as well as you hoped for).
Subject to confirming that any retailer or manufacturer on the list is completely transparent ( see this article) and to making sure that any mattress you are considering meets your specific criteria and the quality/value guidelines here, some options or possibilities I’m aware of in and around the general Ft Lauderdale / Miami / Palm Beach Gardens areas are listed in post #2 here.
If you’re considering purchasing online then a good place to start would be to use the experience and expertise of the members listed in post #21 here who are all very experienced and knowledgeable and specialize in providing the type of help and guidance on the phone that can help you make good choices. There are a wide range of memory foam options included in the choices there and I believe that all of them compete well with the best in the industry in terms of their quality, value, service, and transparency.
When you can’t test a mattress in person then the most reliable source of guidance is always a more detailed phone conversation with a knowledgeable and experienced retailer or manufacturer that has your best interests at heart and who can help “talk you through” the specifics of their mattresses and the properties and “feel” of the materials they are using (fast or slow response, resilience, firmness etc.) and the options they have available that may be the best “match” for you based on the information you provide them, any local testing you have done or mattresses you have slept on and liked or other mattresses you are considering that they are familiar with, and the “averages” of other customers that are similar to you. They will know more about “matching” their specific mattress designs, options, and firmness levels to different body types, sleeping positions, and preferences (or to other mattresses that they are familiar with) than anyone else.
Their detailed knowledge of their mattresses and how they fit with different body types and sleeping positions along with your feedback from local testing, a customer base of many people that they can use as reference points, and any exchange, return, or any options they have available to customize a mattress after a purchase can help lower the risk of an online purchase. These online retailers or manufacturers can also be a good “value reference” for local purchases to make sure that if you are paying a “premium” for a local purchase (in exchange for the kind of “in person” guidance, service, and value that comes with dealing with a local retailer that can help you make more “accurate” choices that you have tested in person) is not too high.
In addition to the online members of the site listed in the previous membership link … post #12 here has some of the better online memory foam options for those who are looking for a memory foam mattress.
Phoenix