  We investigate the pure dephasing of a Josephson qubit due to the spectral
diffusion of two-level systems that are close to resonance with the qubit. We
identify the parameter regime in which this pure dephasing rate can be of the
order of the energy relaxation rate and, thus, the relation $T_2 = 2 T_1$ is
violated for the qubit. This regime is reached if the dynamics of the thermal
TLSs responsible for the spectral diffusion is sufficiently slower than the
energy relaxation of the qubit. By adding periodic bias modulating the qubit
frequency or TLS excitation energies we show that this pure dephasing mechanism
can be mitigated, allowing enhancement of superconducting qubits coherence
time. Mitigating pure dephasing, even if it is subdominant, is of special
significance in view of recent suggestions for converting the dominant
relaxation process ($T_1$) into erasure errors, leaving pure dephasing as the
bottleneck for efficient quantum computation.
