cosmic-microwave-background - 2.7 degrees but does the Big Bang care? The Big Bang is a proposed answer to the beginning of the universe, and the Cosmic Microwave Background or CMB is a central part of the evidence for it.

Now suppose the measurement returned a different value than 2.7K.

What if it were 0.27K or 27K?  Would the Big Bang theory still work with a few minor adjustments or is the specific temperature of 2.7K critical evidence that the theory works?

Edit: I have removed a paragraph which was not mine asking about small changes in temperature. It is a fundamentally different question. I am more interested in how firm the evidence for the BB theory is, not its details. At least for this question. 
However if uhoh who seems to have done the editing thinks it a fair question, I would certainly welcome it - in another post.
The expansion of the universe is described by a scale factor that we normally call $a(t)$. We take the scale factor to be equal to one right now, so if $a(t)$ is less than one that means the universe has contracted while if $a(t)$ is greater than one the universe has expanded.

General relativity allows us to calculate how the scale factor depends on time. If you're interested I do this in the question How does the Hubble parameter change with the age of the universe? on the Physics SE. This is what the results look like:



The point of this is that the CMB temperature is directly related to the scale factor so if we can calculate how the scale factor has changed with time then we can calculate the current CMB temperature.

But the way the scale factor changes with time depends on the density of matter and energy in the universe. Specifically it depends on how much light, matter (normal and dark) and dark energy is present. The matter/energy density could in principle have any values, that is there is no fundamental principle in physics that specifies what the densities of the of the various types of matter and energy have to be.

Our current best measurements of the matter/energy density come from the Planck experiment, and they suggest the total density has a value close to the critical density. This density gives us the form of the scale factor that I drew above, and it gives us the CMB temperature of 2.7K. If the density were lower than critical we would get a lower CMB temperature and if the density were higher than critical we get a higher CMB temperature. So the answer to your question is that the CMB temperature is not uniquely determined and it could be higher or lower. Ultimately the CMB temperature is an experimental parameter that we have to determine by making experimental measurements.

But we should note that there are excellent reasons why we would expect the matter/energy density to be equal to the critical density, so we would expect the CMB temperature to be what we observe. If the CMB temperature were different by a factor of ten that would be hard to explain theoretically.

Finally we should note that there are other independent constraints of the form of the scale factor function. Rob Jeffries describes some of these in his answer so I won't go through them again here, but for example the ratio of hydrogen to helium depends on how quickly $a(t)$ increased after the Big Bang. If the scale factor had increased significantly faster or more slowly it would have changed the hydrogen to helium ratio and we'd be able to measure this change.