Is there an equation for jet engine efficiency as a function of turbine inlet temperature?
The jet engine efficiency is unfortunately more complicated than just a one-to-one function between static Turbine Inlet Temperature and efficiency. Thermodynamic efficiency of a turbine engine is defined as the useful generated power extracted from the chemical energy added by the fuel. 

The following is extracted from a paper format uni book on aircraft gas turbines.



Station 0 is for the engine inlet, the other station numbers are:


Compressor inlet.
Combustion chamber inlet.
Turbine inlet.
Turbine outlet.
Engine exhaust.


Energy IN

The heat flow $\dot{Q}$ added to the engine is:

$$ \dot{Q} = \dot{m} \cdot c_{pg} \cdot (T_{3t} - T_{2t}) \tag{1}$$
With $\dot{m}$ = mass flow through engine, $c_{pg}$ = gas constant, $T_{3t}$ = total temperature at turbine inlet. Total temperature is the temperature reached when a gas flow is compressed isentropically, measured in the stagnation point, and defined as $$T_t = T + v^2/(2 * C_p) \tag{2}$$

So energy IN is a function of:


Static Turbine Inlet Temperature
total mass flow
gas flow velocity at the turbine inlet.


Useful power OUT

The power delivered by the gas generator is:

$$P_{gg} = \dot{}m \cdot c_{pg} \cdot T_{4t} \left[ 1 - {\left(\frac{p_0}{p_{4t}} \right)}^{\frac{\kappa_g - 1}{\kappa_g}} \right] \tag{3}$$

with 


$T_{4t}$ = stagnation temperature at  the turbine outlet.
$p_0$ = static pressure at engine inlet, a function of air density and airspeed.
$p_{4t}$ = stagnation pressure at the turbine outlet, which depends on how much energy the turbine has extracted from the gas flow.


Efficiency

If we could vary the $T_{3t}$ only and keep the other variables constant, we could indeed get to the sort of function sought after - but we cannot. Increasing T.I.T creates more thrust, accelerates the aircraft, increases inlet pressure, increases turbine outlet pressure etc.

From Gas Turbine Theory by Saravanamuttoo/Rogers/Cohen, page 81:


  The large number of variables involved make it impracticable to derive algebraic expressions for the specific output and efficiency of real cycles.