  The spins of a number of supermassive and stellar-mass black holes have been
measured based on detections of thermal continuum emission and relativistically
broadened iron lines in their x-ray spectra. Likewise, quasiperiodic
variability has been observed in several sources. Such measurements commonly
make the assumption that black holes are described by the Kerr metric, which
according to the no-hair theorem characterizes black holes uniquely in terms of
their masses and spins. This fundamental property of black holes can be tested
observationally by measuring potential deviations from the Kerr metric
introduced by a parametrically deformed Kerr-like spacetime. Thermal spectra,
iron lines, and variability have already been studied extensively in several
such metrics, which usually depend on only one particular type of deviation or
contain unphysical regions outside of the compact object. In this paper, I
study these x-ray probes in the background of a new Kerr-like metric which
depends on four independent deviation functions and is free of pathological
regions outside of the event horizon. I show that the observed signals depend
significantly on primarily two types of deviations and that the strong
correlation between the spin and the deviation parameters found previously in
other Kerr-like metrics is partially broken for rapidly spinning black holes.
This suggests that high-spin sources are the best candidates for tests of the
no-hair theorem with x-rays and I obtain first constraints on such deviations
from the stellar-mass black hole Cygnus X-1.
