High sensitivity, fast response time and strong light absorption are the most important metrics for infrared sensing and imaging. The trade-off between these characteristics remains the primary challenge in bolometry. Graphene with its unique combination of a record small electronic heat capacity and a weak electron-phonon coupling has emerged as a sensitive bolometric medium that allows for high intrinsic bandwidths1-3. Moreover, the material's light absorption can be enhanced to near unity by integration into photonic structures. Here, we introduce an integrated hot-electron bolometer based on Johnson noise readout of electrons in ultra-clean hexagonal-boron-nitride-encapsulated graphene, which is critically coupled to incident radiation through a photonic nanocavity with Q = 900. The device operates at telecom wavelengths and shows an enhanced bolometric response at charge neutrality. At 5 K, we obtain a noise equivalent power of about 10 pW Hz(-1/2), a record fast thermal relaxation time, < 35 ps, and an improved light absorption. However the device can operate even above 300 K with reduced sensitivity. We work out the performance mechanisms and limits of the graphene bolometer and give important insights towards the potential development of practical applications.