The thermometer is in something (air, water, let's say water to
make the following easier to write) that has a temperature. It
has a particular temperature; the temperature of that water is
not the same as the temperature of anything else. That
particular temperature is a reality and we can talk of accurate
portrayal of it as "real". The thermometer is not reporting it
accurately, so "the thermometer does not give the real
temperature" is correct. We can also use temperature as an
abstract noun, to talk about the concept of temperature more
widely. Here though, there isn't a specific "real temperature"
to talk of. "Real" with abstract nouns contrasts a view of it we
hold to be genuine with one we don't ("Real love"). Saying it
doesn't measure "real temperature" would make sense if we had
some theory as to why it's working on a completely different
concept of "temperature" to that we hold to be "real". To
express the same thing with an abstract use of temperature we'd
need to avoid real and use something like