Several transiting hot Jupiters orbit relatively inactive main-sequence stars. For some of those, the logRHK activity parameter lies below the basal level (-5.1). Two explanations have been proposed so far: (i) the planet affects the stellar dynamo, (ii) the logRHK measurements are biased by extrinsic absorption, either by the interstellar medium (ISM) or by material local to the system. We present here Hubble Space Telescope/COS far-UV spectra of WASP-13, which hosts an inflated hot Jupiter and has a measured logRHK value (-5.26), well below the basal level. From the stars spectral energy distribution we obtain an extinction E(B-V) = 0.045+/-0.025 mag and a distance d = 232+/-8 pc. We detect at >4 sigma lines belonging to three different ionization states of carbon (C1, C2, and C4) and the Si4 doublet at ~3 sigma. Using far-UV spectra of nearby early G-type stars of known age, we derive a C4/C1 flux ratio-age relation, from which we estimate WASP-13s age to be 5.1+/-2.0 Gyr. We rescale the solar irradiance reference spectrum to match the flux of the C4 1548 doublet. By integrating the rescaled solar spectrum, we obtain an XUV flux at 1 AU of 5.4 erg s^-1 cm^-2. We use a detailed model of the planets upper atmosphere, deriving a mass-loss rate of 1.5x10^11 g s^-1. Despite the low logRHK value, the star shows a far-UV spectrum typical of middle-aged solar-type stars, pointing toward the presence of significant extrinsic absorption. The analysis of a high-resolution spectrum of the Ca2H&K lines indicates that the ISM absorption could be the origin of the low logRHK value. Nevertheless, the large uncertainty in the Ca2 ISM abundance does not allow us to firmly exclude the presence of circumstellar gas.
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