a series of polynomial expansions of the ephemeris that predict the barycentered phase of each photon detected. The length of each data chunk was chosen according to the length of each [[*RXTE*]{}]{} observation (Obs-Id).

We then split each pulse profile into a fundamental (at $\nu$) and a first overtone (at $2\nu$) using standard $\chi^{2}$ fits. For a detailed description of the method used for measuring the pulse time of arrivals (TOAs) we refer to @pat09. A set of pulse phase residuals was then obtained by subtracting a Keplerian circular orbit and constant pulse frequency model, using the ephemeris of @har09.

[[*Swift*]{}-XRT]{} observations
--------------------------------

During the 2005 and 2008 outbursts, the [[*Swift*]{} X-ray Telescope]{} ([[*Swift*]{}-XRT]{}) observed the re-flaring state of J1808. We refer to @cam08 for the study of the [[*Swift*]{}-XRT]{} 2005 observations. Here we consider 10 pointed observations (see Table 1) covering a total of $\sim 15$ ks that were taken during the 2008 outburst (MJD$\,54757$–$54778$) and were reduced by using the XRT pipeline (v. 0.12.0). Each observation lasted between 1 and 3 ks. The data were collected in photon counting (PC) mode, except for observation 00030034041 which was taken in windowed timing (WT) mode.

We extracted source and background 