Rotating Black Holes in Horndeski Gravity: Thermodynamic and Gravitational Lensing

The lack of rotating black holes, typically found in nature, hinders testing modified gravity from astrophysical observations. We present the axially symmetric counterpart of an existing spherical hairy black hole in Horndeski gravity having an additional deviation parameter $Q$, which encompasses the Kerr black hole as a particular case ($Q=0$). We investigate the effect of Horndeski parameter $Q$ on the rotating black holes' geometry and analytically deduce the gravitational deflection angle of light in the weak-field limit. For the S2 source star, the deflection angle for the Sgr A* model of the rotating Horndeski gravity black hole for both prograde and retrograde photons is larger than the Kerr black hole values. We show how parameter $Q$ could be constrained by astrophysical implications of the lensing of this object. The thermodynamic quantities, Komar mass, and Komar angular momentum gets corrected by the parameter $Q$, but the Smarr relation $M_{\text{eff}}=2ST+2\Omega J_{\text{eff}}$ still holds at the event horizon.