I began jogging when Forrest Gump did, I sold sneakers for 3 years, and my wife and I hiked to the bottom of the grand canyon and back within one day, on our honeymoon - oh, I was 6 weeks post-ACL surgery.
Here are a few thoughts.
Coolness comes from nylon fabric versus leather, and from lower top versus higher top.
Stability comes from higher top, "cup" sole, and firm sole. A steel piece in a boot helps give stability. A "cup" sole is a sole that is not totally below the foot, but curves up around the foot. A running shoe has no cup sole. A cross trainer has something of a cup usually at the heel, the outside, and maybe at the toe.
When you are hiking over rough terrain, and especially if you are carrying gear, a stable shoe helps prevent injury.
Also, when setting up a stance for shooting, you will benefit from more stability than a running shoe. Golfers, bowlers, and disc golfers actually prefer a shoe with some stability for their very specific motion. They don't want to be wobbling around.
The part of the shoe above the sole is the 'upper.' Running shoes have the nylon fabric upper to be flexible and cool. Cross-trainers usually are meant to be OK for some tennis, some hiking, some basketball - so they need a leather upper, which is more supportive than the nylon fabric of a running shoe, but they use a leather that is softer than a tennis shoe or a basketball shoe.
Also, the companies try to perforate the leather to be cooler- over the long run this makes the leather stretch out more easily. It is getting obvious that a cross-trainer is quite a compromise. I have run 4-5 miles in a pair, but it was much more of a burden than in running shoes. (I have run a marathon, also - in running shoes.)
There are trail-running shoes, but those are usually just a tough running shoe with no 'cup' to the sole. They have a rougher tread, exactly like an off-road tire versus a z-rated tire.
A boot is just awesome. A cooler boot is even more awesome. Look for boots with the nylon fabric worked into the design on the side, and on the tongue. A tongue can be leather or nylon fabric, or a combo of these.
There are army boots like this, plus hi-tek, a british hiking shoe company, is well-known for their classic hiking boot design with "mesh" / nylon fabric on the side.
So, you need to judge by terrain what you can get away with.
If you are hiking in any running shoe, to be cooler, new balance is probably the best thing going.
Nike has a line of shoes and garments called "ACG" all-conditions gear - my ACG boots and running shoes were not cool. The ACG running shoes were for cold, wet running days on decent surface, and ACG boots for light hiking.
Nike has had some great, stable mid-cut cross-trainers, that are not too hot - a best example might be the Nike Air Trainer SC II.
Another option would be a football or mid-cut baseball cleat with the nylon mesh fabric in the design. I play ultimate Frisbee in cleats all the time, so cleats are no unusual-feeling to me.
Finally, in my opinion, I stay more comfortable with two pairs of socks in my hiking boots. I think more air can circulate in all of that cotton. Others disagree about coolness and comfort of two-pairs versus one pair of socks. Some like wool, I guess because the wooli-ness allows more circulation.
