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Graphical abstract The phase diagram in plane of temperature and magnetic field for the spin-1/2 kagome Heisenberg antiferromagnet is theoretically obtained for the first time. It is comprised of five phases: the gapless quantum spin liquid (QSL) phase, the algebraic paramagnetic liquid (APL) phase that possesses both paramagnetic properties and algebraic behaviors inherited from the QSL, the field-induced ordered phase, the intermediate (spin canted) phase, and the conventional paramagnetic state. Abstract Quantum fluctuations from frustration can trigger quantum spin liquids (QSLs) at zero temperature. However, it is unclear how thermal fluctuations affect a QSL. We employ state-of-the-art tensor network-based methods to explore the ground state and thermodynamic properties of the spin- 1 / 2 kagomé Heisenberg antiferromagnet (KHA). Its ground state is shown to be consistent with a gapless QSL by observing the absence of zero-magnetization plateau as well as the algebraic behaviors of susceptibility and specific heat at low temperatures, respectively. We show that there exists an algebraic paramagnetic liquid (APL) that possesses both the paramagnetic properties and the algebraic behaviors inherited from the QSL. The APL is induced under the interplay between quantum fluctuations from geometrical frustration and thermal fluctuations. By studying the temperature-dependent behaviors of specific heat and magnetic susceptibility, a finite-temperature phase diagram in a magnetic field is suggested, where various phases are identified. This present study gains useful insight into the thermodynamic properties of the spin-1/2 KHA with or without a magnetic field and is helpful for relevant experimental studies. [ABSTRACT FROM AUTHOR]