To stealth or not to stealth: Thermodynamics of stealth black holes

Using Euclidean methods we investigate the thermodynamics of stealth black hole solutions, which are solutions that geometrically are indistinguishable from those of general relativity, however they are accompanied by a non-trivial additional field that does not back-react to the metric. We find that in general, one can observe shifts in the mass and/or the entropy of such black holes, hence, at least at the thermodynamic level, the stealth solutions may be distinguished by the those of general relativity. We also point out an example, the {\it{bona fide stealth}}, where a non-trivial scalar field can accompany an (A)dS-Schwarzschild black hole without altering the thermodynamic quantities. We point out that thermodynamic discrepancies arise due to the additional boundary terms emanating from the stealth fields, where care should be given in order for the theory at hand to possess a well-defined variational procedure.