Is the t-butyl carbocation more stable than the benzyl carbocation?
I am using a very simplistic quantum chemical approach of the following isodesmic reaction:



I have used Gaussian 16 Rev. A.03 and the DF-B97D3/def2-TZVPP level of theory. The summaries of the calculations are included below.

On this level of theory the depicted reaction has an energy change of $\Delta G = \pu{- 37.1 kJ mol-1}.$ Therefore one could assume that the 2-methylpropan-2-ylium cation is more stable than the phenylmethylium cation. These values were estimated at $T = \pu{298.15 K}$ and $p = \pu{1 atm}.$

   

One certainly can do more calculations, but it is, however, a start.



Calculation summaries 


INFO   :  Found route section:
#P B97D3/Def2TZVPP int(ultrafinegrid) freq geom=check guess=read gfinput 
gfoldprint iop(6/7=3) symmetry(none)
----
calculation details               : RB97D3               phch2_q1/b97d3tzvpp.freq.log
temperature                    (T):              298.150 K                   
pressure                       (p):              1.00000 atm                 
electr. en.                    (E):      -270.5777462860 hartree             
zero-point corr.             (ZPE):            +0.114824 hartree/particle    
thermal corr.                  (U):            +0.120571 hartree/particle    
ther. corr. enthalpy           (H):            +0.121515 hartree/particle    
ther. corr. Gibbs en.          (G):            +0.085614 hartree/particle    
entropy (total)            (S tot):              +75.561 cal/(mol K)         
heat capacity (total)       (Cv t):              +22.578 cal/(mol K)         
==== Next file ====
INFO   :  Found route section:
#P B97D3/Def2TZVPP int(ultrafinegrid) freq geom=check guess=read gfinput 
gfoldprint iop(6/7=3) symmetry(none)
----
calculation details               : RB97D3               phch3_q0/b97d3tzvpp.freq.log
temperature                    (T):              298.150 K                   
pressure                       (p):              1.00000 atm                 
electr. en.                    (E):      -271.4854244270 hartree             
zero-point corr.             (ZPE):            +0.125049 hartree/particle    
thermal corr.                  (U):            +0.131428 hartree/particle    
ther. corr. enthalpy           (H):            +0.132372 hartree/particle    
ther. corr. Gibbs en.          (G):            +0.093359 hartree/particle    
entropy (total)            (S tot):              +82.110 cal/(mol K)         
heat capacity (total)       (Cv t):              +23.831 cal/(mol K)         
==== Next file ====
INFO   :  Found route section:
#P B97D3/Def2TZVPP int(ultrafinegrid) freq geom=check guess=read gfinput 
gfoldprint iop(6/7=3) symmetry(none)
----
calculation details               : RB97D3               t-c4h10_q0/b97d3tzvpp.freq.log
temperature                    (T):              298.150 K                   
pressure                       (p):              1.00000 atm                 
electr. en.                    (E):      -158.4201368210 hartree             
zero-point corr.             (ZPE):            +0.128849 hartree/particle    
thermal corr.                  (U):            +0.134606 hartree/particle    
ther. corr. enthalpy           (H):            +0.135550 hartree/particle    
ther. corr. Gibbs en.          (G):            +0.101200 hartree/particle    
entropy (total)            (S tot):              +72.297 cal/(mol K)         
heat capacity (total)       (Cv t):              +20.521 cal/(mol K)         
==== Next file ====
INFO   :  Found route section:
#P B97D3/Def2TZVPP int(ultrafinegrid) freq geom=check guess=read gfinput 
gfoldprint iop(6/7=3) symmetry(none)
----
calculation details               : RB97D3               t-c4h9_q1/b97d3tzvpp.freq.log
temperature                    (T):              298.150 K                   
pressure                       (p):              1.00000 atm                 
electr. en.                    (E):      -157.5167928240 hartree             
zero-point corr.             (ZPE):            +0.114033 hartree/particle    
thermal corr.                  (U):            +0.120557 hartree/particle    
ther. corr. enthalpy           (H):            +0.121501 hartree/particle    
ther. corr. Gibbs en.          (G):            +0.083653 hartree/particle    
entropy (total)            (S tot):              +79.659 cal/(mol K)         
heat capacity (total)       (Cv t):              +20.965 cal/(mol K)         


(Compiled with tools-for-g09.bash, which somewhat surprisingly works for g16.)

Optimised geometries


14
phch2_q1/b97d3tzvpp.freq.xyz
C          0.08663       -0.19347        0.09384
C          0.05691       -0.11475        1.46208
H          1.02702       -0.19311       -0.45239
H         -0.82919       -0.25883       -0.48899
C         -1.20323       -0.11721        2.16356
C         -1.22180       -0.03857        3.53614
C         -0.00341        0.04387        4.24119
C          1.24453        0.04873        3.58467
C          1.28552       -0.02918        2.21254
H         -2.12409       -0.18167        1.59151
H         -2.16067       -0.03932        4.07940
H         -0.02695        0.10566        5.32621
H          2.15902        0.11360        4.16439
H          2.23039       -0.02773        1.67720

15
phch3_q0/b97d3tzvpp.freq.xyz
C          0.00007        0.00000       -0.04694
C          0.00682       -0.00000        1.46143
H          1.01682       -0.00000       -0.44951
H         -0.51968       -0.88199       -0.43936
H         -0.51968        0.88199       -0.43936
C         -1.19805       -0.00000        2.17854
C         -1.20438       -0.00000        3.57251
C          0.00015        0.00000        4.27958
C          1.20614       -0.00000        3.57927
C          1.20606       -0.00000        2.18275
H         -2.14093        0.00000        1.63572
H         -2.14998        0.00000        4.10835
H         -0.00257        0.00000        5.36594
H          2.14940        0.00000        4.11910
H          2.15082        0.00000        1.64397

14
t-c4h10_q0/b97d3tzvpp.freq.xyz
C          0.00369       -0.00664        0.00284
C         -0.01764        0.03020        1.53527
C          1.44139        0.03012       -0.52791
C         -0.74698       -1.23338       -0.52790
H         -1.04360        0.03406        1.92005
H          0.49077       -0.85020        1.94859
H          0.49182        0.92075        1.92008
H          1.46219        0.03417       -1.62347
H          1.97408        0.92055       -0.17569
H          2.00042       -0.85041       -0.18649
H         -0.76068       -1.24961       -1.62346
H         -0.26410       -2.15775       -0.18616
H         -1.78452       -1.24933       -0.17586
H         -0.51485        0.89146       -0.36388

13
t-c4h9_q1/b97d3tzvpp.freq.xyz
C         -0.00184       -0.00056        0.03904
C          0.01681        1.45949        0.05366
H         -0.46367        1.80346       -0.88071
H          1.00698        1.90257        0.15089
H         -0.66770        1.83073        0.83269
C          1.25179       -0.74789        0.02785
H          1.76765       -0.48991       -0.91708
H          1.93393       -0.35706        0.79738
H          1.13840       -1.82916        0.09679
C         -1.27605       -0.71190        0.03180
H         -1.24998       -1.55839       -0.66820
H         -1.35257       -1.20345        1.02233
H         -2.15046       -0.07921       -0.11824


(g09.chk2xyz does not work for g16.)