QCD has a mass gap. Does heating up QCD to a finite temperature change this fact, or is it a property which is independent of temperature?
The answer is that the existence of a mass gap is indeed temperature dependent. The mass gap is linked to the phenomenon of confinement: quarks and gluons do not appear as free particles, but are bound within hadrons. Such bound states cannot have arbitrarily small energy, but their energy is bounded from below by some scale referred to as the mass gap. The question is now whether confinement is a phenomenon to be observed at all temperature scales, and it turns out that this is not the case. At sufficiently high temperatures (depending also on baryon density) there will be deconfinement and quarks and gluons appear as free entities. This phase is commonly referred to as quark gluon plasma and has been experimentally verified at LHC and RHIC. Speaking of phases, QCD exhibits a diverse structure once one looks at its behaviour depending on two parameters, temperature and baryon density. In this case, one speaks of the phase diagram of QCD, which schematically looks like this (credits to this webpage):
There is much more to the QCD phase diagram than is visible in this picture and a lot of it is subject to current research, but it should give you a first idea of the basic structure one encounters.
To summarize: above a certain temperature, there is no longer a mass gap in QCD.
The answer is yes, it does depends on the temperature as phenomenological models show. Actually there's more than that - there is a shift in the mass of any QCD particle due to connection to heat bath as you can read in hep-ph/9711297v2 and
If you interested to get more information look for QCD sum rules at finite temperature.