Field effect transistors (FETs), such as quantum point contact transistors [1], are an alternative to Coulomb blockade devices (SETs)[2] for fast single charge detection in nanos- tructures due to their smaller impedance. In the perspective of a future quantum electronics based on ballistic electrons [3], or the fast readout of charge qubits, a nanosecond time res- olution is needed that is challenging but should be reachable by using nano-transistors. A well-known realization is a top gated single carbon nanotube [4], which works at high fre- quency [5] (see review in [6]). Ultimate gate coupling and finite density of states in the channel push these nano-transistors close to the quantum limit where gate capacitance C$_{g}$ approaches the quantum capacitance C$_{q}$ [5]. Beside large charge sensitivity, an important limiting factor of resolution is thermal noise from hot electrons which is prominent in low- dimensional conductors due to poor energy relaxation [7, 8]. Effect of dissipation can be investigated by measuring the out-of-equilibrium phonon population at finite bias V$_{d}$ [9] but also by noise thermometry [10]. The nano-transistor geometry offers an opportunity to revisit these hot electron effects using the additional control of electronic transmission. The noise thermometry approach of electronic population relies on assumption of a ther- mal distribution and the Johnson-Nyquist formula S$_{I}$ = 4 g$_{n}$k$_{B}$T$_{e}$ relates the current noise spectral density S$_{I}$ to the electronic temperature T$_{e}$ via a so-called ”noise conductance” g$_{n}$ [11]. In general one has g$_{n}$ = g$_{d}$ , the differential drain conductance. The situation is different in gated semiconductors where an additional contribution arises, at finite bias, associated with transconductance g$_{m}$ = ∂I$_{d}$/∂V$_{g}$ where I$_{d}$ is the drain current and V$_{g}$ the gate voltage. This term depends a priori on geometry, at least for the 3D and 2D transistors [11, 12]. We show here that the situation is different at 1D and that a simple relation exists between g$_{d}$ , g$_{m}$ and g$_{n}$ which only brings in the gate coupling factor β = C$_{g}$/C$_{q}$ . The paper reports on current noise and transconductance of top-gated single nanotube transistors. Measurements were carried at 4K to take advantage of the enhanced noise resolution of cryogenic setups and in a GHz bandwidth to overcome the 1 /f α low frequency noise and benefit from the good AC coupling in the contact impedance. Hot electron effects show up both in the thermal current noise and the width of the transconductance peak at the onset of conduction. Using a 1D nano-transistor model based on scattering theory [13] we obtain a generalized Johnson-Nyquist noise formula in the form, 2 S$_{I}$ = 4 ( g$_{d}$ + g$_{m}$/ 2 β ) k$_{B}$T$_{e}$ , (1)