  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 kagome 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 \textit{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.
