The tetragonal Mo$_5$PB$_2$ compound was recently reported to show superconductivity with a critical temperature up to 9.2 K. In search of evidence for multiple superconducting gaps in Mo$_5$PB$_2$, comprehensive measurements, including magnetic susceptibility, electrical resistivity, heat capacity, and muon-spin rotation and relaxation ($mu$SR) measurements were carried out. Data from both low-temperature superfluid density and electronic specific heat suggest a nodeless superconducting ground state in Mo$_5$PB$_2$. Two superconducting energy gaps $Delta_0$ = 1.02 meV (25%) and 1.49 meV (75%) are required to describe the low-$T$ electronic specific-heat data. The multigap features are clearly evidenced by the field dependence of the electronic specific-heat coefficient and the Gaussian relaxation rate in the superconducting state (i.e., superfluid density), as well as by the temperature dependence of the upper critical field. By combining our extensive experimental results with numerical band-structure calculations, we provide compelling evidence of multigap superconductivity in Mo$_5$PB$_2$.
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