  We investigate the effect of turning on temperature for the charge neutral
phase of two-flavor color superconducting (2SC) dense quark matter in presence
of constant external magnetic field. Within the Nambu-Jona-Lasinio model, by
tuning the diquark coupling strength, we study the interdependent evolution of
the quark Bardeen-Cooper-Schrieffer gap and dynamical mass as functions of
temperature and magnetic field. We find that magnetic field $B \gtrsim 0.02$
GeV$^2$ ($10^{18}$ G) leads to anomalous temperature behavior of the gap in the
gapless 2SC phase (moderately strong coupling), reminiscent of previous results
in the literature found in the limit of weak coupling without magnetic field.
The 2SC gap in the strong coupling regime is abruptly quenched at ultrahigh
magnetic field due to the mismatched Fermi surfaces of up and down quarks
imposed by charge neutrality and oscillation of the gap due to Landau level
quantization. The dynamical quark mass also displays strong oscillation and
magnetic catalysis at high magnetic field, although the latter effect is
tempered by nonzero temperature. We discuss the implications for newly born
compact stars with superconducting quark cores.
