Why is it more energy efficient to keep a certain temperature in my house (instead of stop heating)? Today I got this recommendation

German for:

Savings-tipp of the month: Turn down your heating on leaving the apartment. But never turn it off completely. Room temperature reduction is much more economic than the reheating of a cold room*

It crashes fundamentally with my physics conception: The higher the temperature difference is, the faster is the equalization process. Hence the more I raise the difference the increasingly higher is the loss.
This has made me assume: In any case would it be more economic to stop heating the moment I don't need the heat.
What do I miss out?
 A: I'm not at all convinced that a little condensation (if any) will change the heat flow thru walls significantly.   Not to mention that heat flow thru walls depends a lot on what construction types, what insulation, etc.
Next: it is true that some heat pumps, under some delta-temperature regimes, work far less efficiently, and it really does take less energy to drive the heat pump continuously than to let your house "dive" to 8-12 degrees C.  If you have a heat pump system, study the owner's manual and/or consult with the manufacturer.
In the case hot water supply systems (and  steam heat), leaving the burners off for long periods -- many days, not hours -- can lead to leaks in various connection points as materials age.  Nearly all plumbing "likes" to stay in a small temperature range so that dissimilar thermal expansion coefficients don't cause stresses on the system.  This is a significant concern in older systems, but from a physics standpoint completely separate from the energy cost of letting a house cool.
A: Notice it says "apartment". So we have a set of rooms in a building that has many more rooms in which we do not control the heating.
The advice is comparing the advised behaviour to an hypothetical behaviour where you repeatedly leave the apartment to get very cold and then heat it up "completely" and says leaving the heating on is "more economical". For whom, it does not say.
It could mean that it is more economical for you and also $for~the~whole~building$. This could be true if the hypothetical "heat up completely" operation is very costly for some reason, i.e. if people coming back to very cold rooms tend to overheat their apartment for a long enough time. I don't think that is a rational or dominant behaviour, but maybe it happens where you live.
It could mean it is more economical for the whole building but not for you, because by not turning off the heating, you will decrease heat losses of neighboring apartments which then won't be motivated to use heating more. The cost of you keeping the heating on may be lower than the cost of everybody reacting to you not heating. Whether that's economic for you, depends on how the pay scheme works - if you pay what you consume, then it's not economic for you. If you pay some pre-determined share, it can be.
If you want to really and reliably save (not use) heating energy, don't heat the whole room. Wear more clothes and use infrared lamp to heat only your body and closest environment when and where you need it. This may sound like a regress to past in comfort, but it is surprisingly acceptable (based on my experience).
A: There are a couple of good answers, but I'm going to answer it from a third perspective not related to a specific type of heating used or shared walls/floors/ceiling with neighboring apartments.
Heating a building requires two things: heating the air, and heating the structure itself.  The thermostat is set so that the heating unit will run until the air flow around the thermostat is at an upper bound at or slightly above the desired temperature.  For some types of heating such as forced air the air temperature will actually heat up quite quickly, because while the heating unit is on the air is cycled through a heat exchanger.  For other types of heat such as radiative this is not quite as true because heat transfer is more of a function of distance from the heat source, but generally the air will heat up quickly anyway (although a lot less evenly!) because the thermal mass of the air isn't that high.
So if all there was to it was heating the air, then it would make a lot of sense to simply turn off the heat when you were gone (or turn it just high enough to prevent freezing to prevent damage to pipes, etc.).  However, there is also the thermal mass of the building.  This is greatly dependent on the construction materials used.  A "stick frame" house will not have as much thermal mass as block construction, concrete, brick, etc. largely because there are a lot of air gaps (often filled with insulation).
Whether or not this works out better to using less energy than keeping the temperature constant may seem to depend on how well insulated the building is.  But you have the same insulation whether you are keeping it the same temperature or letting it cool.  So by that logic you are right that the T in calculating heat loss is less in a cooler building.
If we look at the heat capacity of various materials based on density (so we can compare to air), we get 0.231 MJ/m^3K for wood, approximately 2 for both brick and concrete, and only 0.0012 for air.  So from this we can see that the energy required to heat the air itself is small per unit volume, and even wood is only 1/10th that of brick or concrete.
The energy required to heat up the building is defined by Q = CT where Q is the energy required, C is the thermal mass, and T is the temperature difference.  So from this it can be seen that a building with a large thermal mass will require a lot of energy to bring it back up to temperature after it has cooled.  But conversely, it will also take a long time for the building to cool as more energy much be lost per degree temperature drop.  If energy costs vary over the day, thermal mass (if large enough) can be used to shift energy use to times when energy costs less.
So what the physics is telling us (or me, at least) is that keeping the building as cool as possible at all times uses the least amount of energy.  But, given all of the above I believe that the recommendation being given to keep the temperature from going to low is based on the subjective perception that since it takes a long time to heat the building structure this is going to cost more than simply keeping the temperature a bit warmer all the time, ignoring the longer time it took the temperature to drop.
From a practical standpoint, heating up the entire structure may overwork a heating system that is undersized (or not oversized) and which must run continuously for an extended period of time to initially heat the entire structure.  Depending on how the building was constructed and how heat is transfer this could also result in the system cycling more than designed as the air quickly heats up, the heat shuts off, and then the heat starts to conduct and radiate into the walls causing the temperature to fall and the heat to come on again more quickly than normal.  So there could be added costs not directly related to heating energy but related to maintaining the heating system itself.
A: You don't miss out, here is an article with some more information:

Is it cheaper to leave the heating on low all day or turn it on only when I need it?
This is a hotly debated one. According to experts at the Energy Saving Trust, the idea it's cheaper to leave the heating on low all day is a myth. They're clear that having the heating on only when you need it is, in the long run, the best way to save energy, and therefore money. (A timer's best as your thermostat turns your heating on and off to keep your home at the temperature you set.)
The key thing to understand here is that it's all about the total amount of energy required to heat your home.
It's a given that a certain amount of energy is constantly leaking out of your home (how much will depend on how good your insulation is). The Energy Saving Trust says if you're keeping the heating on all day you're losing energy all day, so it's better to heat your home only when you need it.

However, it's not quite that clear-cut. Some specialists disagree – and argue you should keep the heating on constantly for an entirely different reason.
They advocate keeping the heating on low all day, turning all radiator valves up to the max and the boiler down to the minimum, and say the problem with turning the heating on and off is that every time it's turned off, condensation collects within the walls. This condensation can help conduct heat outside the home, they say – meaning you leak heat more quickly and so will use more energy as a result.

The linked article says:

I note that in your column you recommend running the boiler on the “24 hour” setting at low temperature during cold spells. Why is this please? And would this not imply that boilers are conventionally run too hot?
The point of the exercise is to try to prevent condensation, because when condensation occurs, every time the heating system starts up, it first has to use energy to evaporate the condensed water, before the energy can go towards heating the house. Remember that condensation is not something that occurs only on window panes – it forms chiefly within the fabric of the walls, thus lowering their thermal insulation value.
That’s why I suggest aiming to keep the radiators constantly warm, rather than cycling them from cold to very hot and back again. Think of the motoring analogy, where everyone agrees it is more fuel-efficient to drive at a steady speed rather than continually stopping and starting.

So your instinct is right, but the question is whether the condensation effect is large enough to make a difference. That will no doubt depend on the specifics of your room (inside temperature, outside temperature, type of isolation, etc.)
