Have you cleaned out your dryer vent all the way to the exterior?

On which cycles is the dryer doing this?

Is this a condenser dryer, or an older vented dryer model?

Dryers generally are pretty simple machines aside from all the electronics that seem to be on everything. My first guess would be a bad thermostat based on that description.

Yeah, a bad thermostat, or a physical reason for it to kick in. Like the dryer duct is blocked. People forget to clean those out regularly. The ones that go straight up through the roof are the worst about that.

" Is it something you have to replace or clean?"

Yes. I'd try cleaning first because it might just have gotten gunked up by the prior owners using fabric softeners (a/k/a dryer sheets). Those can leave waxy residues. For how to get to the thermostat, google with a search string like like "youtube + bosch dryer + thermostat." I've found that RepairClinic.com often has pretty helpful DIY videos.

I'm not sure this is something you should be tackling. Or did I understand you correctly as saying "I'm not sure" when Zenhar asked you if this is a condenser or vented dryer?

Maybe it would help to re-ask the question a little differently. When you say that the dryer "just blows cold air" --- where is the air coming out? Is it blowing out a round 4" diameter opening in the back of the machine that is meant to connect with ducting to the outside of your home/building/unit? That would be a "vented" dryer. Or, does this dryer have a couple of small access panels/doors at the bottom front of the machine and no piping in the rear except a thin drain hose? That would be a ventless (a/k/a condenser) model.

So, when you say you "cleaned the filter," were you speaking about the lint catcher that pulls out from the bottom of the door frame? On recent Bosch dryers, there will be moisture sensors on that filter frame as well. Have you cleaned them, too? If not, they may be gunked up and prematurely switching your machine into cool-down mode. The owner's manual has instructions on how to clean them. (If you do not have the manual, find your dryer's actual model number --- it will be on a sticker or placard and will read something like WTG86400 --- then google it to find and download a pdf of the manual.)

If you have one of the ventless/condenser models of Bosch dryer, you also have to clean out the heat exchanger. On more recent models, that will be behind the access panel at bottom left-hand side of the front of the washer. Again, the manual will explain how to extract, wash it out and reinsert it.

Likewise, if you have a ventless/consdenser model, you also need to check and clean the dryer's drain hose because that can get clogged with lint, too.

Another question: is your dryer pushed flat against the back wall and maybe pushed under a counter between cabinets? IIRC, the install instructions for the most recent Bosch condenser dryer models require something like 2 or 3 inches clearance between the rear of the dryer and the back wall, apparently for airflow. Again, check your dryer's manual to see if it requires rear clearance/airspace.

If you have a an older vented Bosch dryer, then I would focus on being sure the vent is entirely clear. That's not only the vent port on the back of the machine but the whole of the vent ducting to the outside. If you have a shop vac, you can try running the hose up through the vent ducting to the exterior flap. Or, hardware stores sell extendable duct brushes. Also, check the exterior flap assembly to make sure there isn't lint blocking that. (Some of them are installed with grates or screens at the end which do keep out critters but also collect lint that can block the airflow to the extent that it trips the overheat circuits.)

A couple more questions: are you running a timed dry cycle or running an automatic program? Did you run your tests with wet clothes in the dryer or are you running it empty to test it? In my minimal limited experience using a Bosch dryer, it would not run heat for very long when empty. Also, on the one I used, the automatic cycle seemed to be very conservative about protecting against over-drying. As soon as something was dry, it seemed to complete an automatic cycle even though other items in the load might still be damp. Dunno if that's true of all Bosch models or if that is what's happening with your loads, but thought I'd ask about that, too.