We examine the lepton-specific 2HDM as a solution of muon g − 2 anomaly under various theoretical and experimental constraints, especially the direct search limits from the LHC and the requirement of a strong first-order phase transition in the early universe. We find that the muon g-2 anomaly can be explained in the region of 32 < tan β < 80, 10 GeV < m A < 65 GeV, 260GeV < m H < 620 GeV and 180 GeV < m H ± < 620 GeV after imposing the joint constraints from the theory, the precision electroweak data, the 125 GeV Higgs data, the leptonic/semi-hadronic τ decays, the leptonic Z decays and Br(B s → µ + µ − ). The direct searches from the h → AA channels can impose stringent upper limits on Br(h → AA) and the multi-lepton event searches can sizably reduce the allowed region of m A and tan β (10 GeV < m A < 44 GeV and 32 < tan β < 60). Finally, we find that the model can produce a strong first-order phase transition in the region of 14 GeV < m A < 25 GeV, 310 GeV < m H < 355 GeV and 250 GeV < m H ± < 295 GeV, allowed by the explanation of the muon g − 2 anomaly.