  We study transport properties and the charge quantization phenomenon in a
small metallic island connected to the leads through two quantum point contacts
(QPCs). The linear conductance is calculated perturbatively with respect to
weak tunneling and weak backscattering at QPCs as a function of the temperature
$T$ and gate voltage. The conductance shows Coulomb blockade (CB) oscillations
as a function of the gate voltage that decay with the temperature as a result
of thermally activated fluctuations of the charge in the island. The regimes of
quantum, $T \ll E_C$, and thermal, $T \gg E_C$, fluctuations are considered,
where $E_C$ is the charging energy of an isolated island. Our predictions for
CB oscillations in the quantum regime coincide with previous findings in [A.
Furusaki and K. A. Matveev, Phys. Rev. B {\bf 52}, 16676 (1995)]. In the
thermal regime the visibility of Coulomb blockade oscillations decays with the
temperature as $\sqrt{T/E_C}\exp(-\pi^2T/E_C)$, where the exponential
dependence originates from the thermal averaging over the instant charge
fluctuations, while the prefactor has a quantum origin. This dependence does
not depend on the strength of couplings to the leads. The differential
capacitance, calculated in the case of a single tunnel junction, shows the same
exponential decay, however the prefactor is linear in the temperature. This
difference can be attributed to the non-locality of the quantum effects. Our
results agree with the recent experiment [S. Jezouin {\em et al}., Nature {\bf
536}, 58 (2016)] in the whole range of the parameter $T/E_C$.
