  Using the Matsubara Green function formalism we calculate the temperature
dependence of spin Hall and spin Nernst conductivities of a two-dimensional
electron gas with Rashba spin-orbit interaction in the linear response regime.
In the case of spin Nernst effect we also include the contribution from
spin-resolved orbital magnetization, which assures correct behavior of the spin
Nernst conductivity in the zero-temperature limit. Analytical formulas for the
spin Hall and spin Nernst conductivities are derived in some specific
situations. Using the Ioffe-Regel localization criterion, we have also
estimated the range of parameters where the calculated results for the spin
Hall and spin Nernst conductivities are applicable. Analytical results show
that the vertex correction totally suppresses the spin Hall conductivity at
arbitrary temperature. The spin Nernst conductivity, in turn, vanishes at $T=0$
when the orbital contribution is taken into account, but generally is nonzero
at finite temperatures.
