Let me suggest a totally different approach: Why not work with the cool conditions instead of against?
You could let the dough proof for a long time, e.g. overnight in the fridge.
This allows for a lot less yeast and hence a less yeasty taste, which is usually desired.
Also, more complex flavors develop during long proofing times.
(There is a reason french baguette may wait for two days before baking.)
For a start, aim for about 5% yeast 1 and use cool instead of lukewarm liquid.
You might have to adjust your attitude, because this requires some planning ahead, but gives you more degrees of freedom on the other hand:
Fresh bread / cake in the morning without having to get up before dawn and more tolerance with regard to proofing / timing - the dough can handle an extra hour or two in the fridge easily.
(Especially nice if you have a baby or a crazy schedule...)
You could also dump the dough in cold water and proof it there.
As crazy as it sounds, it works.
See more here .
If you'd rather stick with warm conditions, I'd aim for 30°C for optimum (=quickest) activity.
Yeast starts to die at about 45°C, completely dead at 55°C.
Also, warmer dough tends to proof unevenly and have a "flat" taste (can't find a better word).
1 fresh yeast, percentage based on flour weight.
Adjust dry yeast accordingly: fresh to dry conversion rate is 1:3, so use 2% dry yeast.
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50C (122F) would be a very high proofing temperature.
The thermal death point of yeast is 55C, and you'll definitely hit a point of diminishing returns if you get too hot (most likely, you will have really rapid proofing on the outside of the loaf and an underproofed "core").
I would recommend setting your oven to the lowest temperature, and then once it feels perceptibly warm, turn it off and use it to proof.
If you're doing a long proof, you can turn it back on for a minute or two every so often, but I wouldn't ever let it actually get up to temperature.
cooking.stackexchange.com/users/16863/SourDoh
In the winter, I usually get fine results proofing in a bowl with a second bowl inverted on top of it, and then putting the whole thing in the oven, turned off, and just the light on.
The light bulb usually produces enough heat to keep the inside of my oven at about 90˚F (32˚C?), and that gives me a good rise.
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Rise from chilled @ 32C and keep rising until around 80-85% of desired height has been achieved.
Return back to the chiller, it will continue to rise for a short time, keep in the chiller until dough is firm again and stable enough to take the weight of toppings.
cooking.stackexchange.com/users/48366/Simon
A commonly quoted temperature to never exceed with warm ingredients or proofing environments is 43°C.
If using an oven, cover your dough container, check oven with a thermometer beforehand, and be aware of radiated heat effects from the elements themselves.
cooking.stackexchange.com/users/35312/rackandboneman
I use the pull-out warming tray on my oven set on its lowest temperature.
I've never actually checked the temperature, but its rather cool.
I can easily touch the bottom of the drawer without any discomfort.
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I've done this a few times, kinda - we have a baking stone, and I'll preheat that, make sure it's cool enough to touch, and set the dough on the warmed stone.
It works pretty well to give a warm environment, the warmth lasts because the closed oven contains heat, and there's little to no risk of overheating the yeast.
I have both gone with regular preheating for ~5min, and the lowest-temperature-preheating for a bit longer, the former is quicker to heat but more likely to overshoot (or needing time to even the heat out), the latter slower and more controlled.
Another potential alternative, depending on your oven, might be to turn the oven on (bake setting, usually) but keep the temperature set to zero.
In our oven, that turns on the light, the machinery is humming, and the oven gets kinda warm - not hot, but just warm, preparing to heat but not doing so.
I've used this to dehydrate things before, because I can leave it going for quite some time without needing to re-check or re-warm.
It wouldn't work for me with bread dough, since with aforementioned baking stone it would take longer to warm up than it would for the dough to actually rise (since especially the bottom of the dough, against the stone, will rise very differently from the top) .
But if you don't have a baking stone it may work very well for you.
cooking.stackexchange.com/users/47365/Megha
During wintertime it is often cold enough in my apartment that proofing takes much longer than in summer. I'd like to use my oven to create a warmer environment, in which the dough would rise faster. I can go 30-50 degrees celsius, the question is what temperature is safe and at what temperature will I actually start baking the dough. Is 50 degrees celsius still ok or not?
Using low oven temperature to accelerate proofing
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<bread><dough><oven><rising><proofing>
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Yeast starts to die at about 45°C, and is completely dead at 55°C, so even an oven on low might easily be too warm to ensure consistent rising. You could try turning your oven to the lowest setting until it feels perceptibly warm, then turning it off and adding the (covered) dough. A slightly warmed baking stone could also work, or even leaving the pilot light on. When using an oven, check the temperature with a thermometer and be aware of radiated heat from the elements. Alternatively, you could work with the cooler temperatures and allow the dough to rise overnight in the fridge. The oven can be used in many ways to create a suitable warm environment for rising bread dough but care must be taken not to let it get too hot and kill the yeast. Moreover, it is possible to rise bread dough in cold conditions, it just takes longer.
It is possible to rise bread dough in cold conditions. It takes longer but it works. There are a number of oven settings recommended for rising bread dough. The most important thing is to make sure it does not get too warm and kill the yeast.  Yeast dies at 55°C so bread dough needs to rise at a much lower temperature.