  We study convergence to the invariant measure for a class of semilinear
stochastic evolution equations driven by L\'evy noise, including the case of
cylindrical noise. For a certain class of equations we prove the exponential
rate of convergence in the norm of total variation. Our general result is
applied to a number of specific equations driven by cylindrical symmetric
$\alpha$-stable noise and/or cylindrical Wiener noise. We also consider the
case of a "singular" Wiener process with unbounded covariance operator. In
particular, in the equation with diagonal pure $\alpha$-stable cylindrical
noise introduced by Priola and Zabczyk we generalize results in [12]. In the
proof we use an idea of Maslowski and Seidler from [10].
