János Pintz considered such questions recently, see his preprints here and here. In particular, under a weak form of the Elliot-Halberstam conjecture there is an integer $d>0$ such that there are arbitrary long arithmetic progressions of primes $p$ such that $p+d$ is the next prime. Assuming the full conjecture one can take $d\leq 16$, while under a natural strengthening of it one can take any even number $d>0$.

János Pintz considered such questions recently, see his preprints here and here. In particular, under a weak form of the Elliot-Halberstam conjecture there is an integer $d>0$ such that there are arbitrary long arithmetic progressions of primes $p$ such that $p+d$ is the next prime. Assuming the full conjecture one can take $d\leq 16$, while under a natural strengthening of it one can take any even number $d>0$.

János Pintz considered such questions recently, see his preprints here and here. In particular, under a weak form of the Elliot-Halberstam conjecture there is $d>0$ such that there are arbitrary long arithmetic progressions of primes $p$ such that $p+d$ is the next prime. Assuming the full conjecture one can take $d\leq 16$, while under a natural strengthening of it one can take any even number $d>0$.

János Pintz considered such questions recently, see his preprints here and here. In particular, under a weak form of the Elliot-Halberstam conjecture there is an integer $d>0$ such that there are arbitrary long arithmetic progressions of primes $p$ such that $p+d$ is the next prime. Assuming the full conjecture one can take $d\leq 16$, while under a natural strengthening of it one can take any even number $d>0$.