  We present a detailed study of the X-ray emission from PSR B1055-52 using
XMM-Newton observations from 2019 and 2000. The phase-integrated X-ray emission
from this pulsar is poorly described by existing neutron star atmosphere
models. Instead, we confirm that, similar to other middle-aged pulsars, the
best-fitting spectral model consists of two blackbody components, with
substantially different temperatures and emitting areas, and a nonthermal
component characterized by a power law. Our phase-resolved X-ray spectral
analysis using this three-component model reveals variations in the thermal
emission parameters with the pulsar's rotational phase. These variations
suggest a nonuniform temperature distribution across the neutron star's
surface, including the cold thermal component and probable hot spot(s). Such a
temperature distribution can be caused by external and internal heating
processes, likely a combination thereof. We observe very high pulse fractions,
60\%--80\% in the 0.7-1.5, keV range, dominated by the hot blackbody component.
This could be related to temperature non-uniformity and potential beaming
effects in an atmosphere. We find indication of a second hot spot that appears
at lower energies (0.15-0.3, keV) than the first hot spot (0.5-1.5, keV) in the
X-ray light curves, and is offset by about half a rotation period. This finding
aligns with the nearly orthogonal rotator geometry suggested by radio
observations of this interpulse pulsar. If the hot spots are associated with
polar caps, a possible explanation for their temperature asymmetry could be an
offset magnetic dipole and/or an additional toroidal magnetic field component
in the neutron star crust
