  Hawkes process (HP) is a point process with a conditionally dependent
intensity function. This paper defines the tempered fractional Hawkes process
(TFHP) by time-changing the HP with an inverse tempered stable subordinator. We
obtained results that generalize the fractional Hawkes process defined in
Hainaut (2020) to a tempered version which has \textit{semi-heavy tailed}
decay. We derive the mean, the variance, covariance and the governing
fractional difference-differential equations of the TFHP. Additionally, we
introduce the generalized fractional Hawkes process (GFHP) by time-changing the
HP with the inverse L\'evy subordinator. This definition encompasses all
potential (inverse L\'evy) time changes as specific instances. We also explore
the distributional characteristics and the governing difference-differential
equation of the one-dimensional distribution for the GFHP.
